Introduction
The random intersection graph model, denoted by G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) [10 ] . In some respects it resembles the classical G ( n , p ) \mathcal{G}(n,p) G ( n , p ) p p p G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) G ( n , p ) \mathcal{G}(n,p) G ( n , p ) [10 ] focused on threshold functions for the property of containing a copy of a given graph H H H G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) G ( n , p ) \mathcal{G}(n,p) G ( n , p ) [8 , 12 , 21 , 5 ] .
In this article, we focus on the subgraph count in G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) G ( n , p ) \mathcal{G}(n,p) G ( n , p ) [20 ] and complemented by convergence rates in the Wasserstein distance in [1 ] . An analogous result for the more natural Kolmogorov distance was first derived in [19 ] for triangles, and later extended to arbitrary graphs in [17 ] . Recently, many results dealing with other random graph models have appeared; see, for example, [2 , 7 , 11 , 13 , 14 , 15 , 16 , 18 ] .
In the context of the asymptotic distribution of the number of small subgraphs in G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) [24 , 25 ] . For certain related random intersection graph models and in various parameter regimes, normal approximations for the number of triangles [3 ] and for the number of 2 2 2 [4 ] have been established. However, none of the techniques developed in these studies can be adapted to derive conditions for the asymptotic normality of the number of small cliques in G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) [6 ] . However, the distance bounds obtained there are far from optimal. In particular, they do not imply convergence to the normal distribution for m ≥ n 2 m \ge n^2 m ≥ n 2 [8 , 12 , 21 , 5 ] show that the larger the value of m m m G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) G ( n , p ) \mathcal{G}(n,p) G ( n , p ) [6 ] assume that the edge probability tends to zero.
Recently, the authors of this paper have developed a new method that considerably simplifies the analysis of the normal approximation for the number of small subgraphs in random intersection graphs. In [22 ] , an application of this method was demonstrated for the number of edges and a wide range of parameters was considered. In the present paper, we develop new combinatorial tools aimed at extending this approach from [22 ] to the more general setting of arbitrary cliques. Our results cover the full range of parameters typically considered in the context of G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) 1 1 1 [22 ] . Nevertheless, considering larger graphs significantly increases the complexity of the problem, which required a detailed analysis of subgraphs probabilities. We develop techniques for analysing so-called clique covers. These methods allow for the comparison of occurrence probabilities of different subgraphs, which proves crucial for studying the central moments of certain random variables – a necessary step in establishing convergence to the normal distribution. We believe that these results may be of independent interest, as the techniques introduced here can be applied to study the probabilities of other subgraphs in random intersection graphs. Furthermore, the general framework developed in this work may also prove valuable for estimating central moments in other models of random graphs.
In the article we use the following asymptotic notation:
a n = o ( b n ) ⇔ a n / b n → 0 a_n = o(b_n) \Leftrightarrow a_n/b_n \to0 a n = o ( b n ) ⇔ a n / b n → 0 n → ∞ n \to\infty n → ∞
a n = O ( b n ) ⇔ ∣ a n ∣ ≤ C b n a_n = O(b_n) \Leftrightarrow\lvert a_n\rvert\le Cb_n a n = O ( b n ) ⇔ ∣ a n ∣ ≤ C b n C > 0 C > 0 C > 0 n ∈ N n \in\mathbb{N} n ∈ N
a n ≲ b n a_n \lesssim b_n a n ≲ b n b n ≳ a n ⇔ a n = O ( b n ) b_n \gtrsim a_n \Leftrightarrow a_n = O(b_n) b n ≳ a n ⇔ a n = O ( b n ) a n , b n > 0 a_n,b_n > 0 a n , b n > 0
a n ≍ b n ⇔ a n ≲ b n a_n \asymp b_n \Leftrightarrow a_n \lesssim b_n a n ≍ b n ⇔ a n ≲ b n b n ≲ a n b_n \lesssim a_n b n ≲ a n
All constants implicit in the asymptotic notation are functions of the size of the considered graph.
We also use standard notation from graph theory. For a graph G = ( V ( G ) , E ( G ) ) G=(V(G),E(G)) G = ( V ( G ) , E ( G )) V ( G ) V(G) V ( G ) E ( G ) E(G) E ( G ) G ′ ⊆ G G' \subseteq G G ′ ⊆ G G G G V ( G ′ ) ⊆ V ( G ) V(G') \subseteq V(G) V ( G ′ ) ⊆ V ( G ) E ( G ′ ) ⊆ E ( G ) E(G') \subseteq E(G) E ( G ′ ) ⊆ E ( G ) V ′ ⊆ V ( G ) V' \subseteq V(G) V ′ ⊆ V ( G ) E ′ = { e ∈ E ( G ) : e ⊆ V ′ } E'=\{e\in E(G):e\subseteq V'\} E ′ = { e ∈ E ( G ) : e ⊆ V ′ } G [ V ′ ] = ( V ′ , E ′ ) G[V']=(V',E') G [ V ′ ] = ( V ′ , E ′ ) G G G V ′ V' V ′ E ′ ⊆ E ( G ) E'\subseteq E(G) E ′ ⊆ E ( G ) V ′ = { v : ∃ e ∈ E ′ v ∈ e } V'=\{v:\exists e\in E'\ v\in e\} V ′ = { v : ∃ e ∈ E ′ v ∈ e } G [ E ′ ] = ( V ′ , E ′ ) G[E']=(V',E') G [ E ′ ] = ( V ′ , E ′ ) G G G E ′ E' E ′ G − E ′ = ( V ( G ) , E ( G ) ∖ E ′ ) G-E'=(V(G),E(G)\setminus E') G − E ′ = ( V ( G ) , E ( G ) ∖ E ′ ) G ∪ G ′ = ( V ( G ) ∪ V ( G ′ ) , E ( G ) ∪ E ( G ′ ) ) G\cup G'=(V(G)\cup V(G'),E(G)\cup E(G')) G ∪ G ′ = ( V ( G ) ∪ V ( G ′ ) , E ( G ) ∪ E ( G ′ )) G ∩ G ′ = ( V ( G ) ∩ V ( G ′ ) , E ( G ) ∩ E ( G ′ ) ) G\cap G'=(V(G)\cap V(G'),E(G)\cap E(G')) G ∩ G ′ = ( V ( G ) ∩ V ( G ′ ) , E ( G ) ∩ E ( G ′ )) deg G ( v ) = ∣ { e ∈ E ( G ) : v ∈ e } ∣ \deg_G(v)=|\{e\in E(G):v\in e\}| deg G ( v ) = ∣ { e ∈ E ( G ) : v ∈ e } ∣ v v v G G G ≅ \cong ≅
We denote by K i K_i K i i i i K \mathbf K K K x \mathbf K_x K x x x x K r K_r K r r ≥ 2 r\geq2 r ≥ 2 N i N_i N i i i i N 0 N_0 N 0 K 0 K_0 K 0
Main results and discussion
Let us first introduce the random intersection graph model G ( n , m , p ) \mathcal G(n,m,p) G ( n , m , p ) n ∈ N n\in\mathbb N n ∈ N V \mathcal V V ∣ V ∣ = n |\mathcal V|=n ∣ V ∣ = n A = { a 1 , … , a m } \mathcal A=\{a_1,\ldots,a_m\} A = { a 1 , … , a m } m = m ( n ) ∈ N m=m(n)\in\mathbb N m = m ( n ) ∈ N a i a_i a i 1 ≤ i ≤ m 1\leq i\leq m 1 ≤ i ≤ m v ∈ V v\in\mathcal V v ∈ V v v v a i a_i a i p = p ( n ) ∈ ( 0 , 1 ) p=p(n)\in(0,1) p = p ( n ) ∈ ( 0 , 1 ) v v v v ′ v' v ′ G ( n , m , p ) \mathcal G(n,m,p) G ( n , m , p ) p \mathbf p p
p = 1 − ( 1 − p 2 ) m ≍ m p 2 ∧ 1 , (1) \mathbf p=1-(1-p^2)^m\asymp mp^2\wedge1, \tag*{(1)} p = 1 − ( 1 − p 2 ) m ≍ m p 2 ∧ 1 , (1) where the estimate is valid for p < c p<c p < c c ∈ ( 0 , 1 ) c\in(0,1) c ∈ ( 0 , 1 )
We study the random variable X n ( r ) X_n^{(r)} X n ( r ) r ≥ 2 r\geq2 r ≥ 2 r r r G ( n , m , p ) \mathcal G(n,m,p) G ( n , m , p ) K \mathcal K K r r r V \mathcal V V K ∈ K \mathbf K\in\mathcal K K ∈ K I K \mathbb I_{\mathbf K} I K { K ⊆ G ( n , m , p ) } \{\mathbf K\subseteq\mathcal G(n,m,p)\} { K ⊆ G ( n , m , p )} P = E [ I K ] = P ( K ⊆ G ( n , m , p ) ) \mathbf P=\mathbb E[\mathbb I_{\mathbf K}]=\mathbb P(\mathbf K\subseteq\mathcal G(n,m,p)) P = E [ I K ] = P ( K ⊆ G ( n , m , p )) K ∈ K \mathbf K\in\mathcal K K ∈ K
Then
X n ( r ) = ∑ K ∈ K I K , and E [ X n ( r ) ] = ∑ K ∈ K E [ I K ] = ( n r ) P . X_n^{(r)}=\sum_{\mathbf K\in\mathcal K}\mathbb I_{\mathbf K},\qquad\text{and}\qquad\mathbb E[X_n^{(r)}]=\sum_{\mathbf K\in\mathcal K}\mathbb E[\mathbb I_{\mathbf K}]=\binom{n}{r}\mathbf P. X n ( r ) = K ∈ K ∑ I K , and E [ X n ( r ) ] = K ∈ K ∑ E [ I K ] = ( r n ) P . Note that estimates of P \mathbf P P m m m p p p p \mathbf p p [10 ] , and are recalled below in Lemma 4.5 . Our main goal is to examine the asymptotic normality of the standardized random variable
X ~ n ( r ) : = X n ( r ) − E [ X n ( r ) ] Var [ X n ( r ) ] . \widetilde{X}_n^{(r)}:=\frac{X_n^{(r)}-\mathbb E[X_n^{(r)}]}{\sqrt{\operatorname{Var}[X_n^{(r)}]}}. X n ( r ) := Var [ X n ( r ) ] X n ( r ) − E [ X n ( r ) ] . and to establish bounds on its convergence rates to a normally distributed random variable N ∼ N ( 0 , 1 ) \mathcal N\sim\mathcal N(0,1) N ∼ N ( 0 , 1 )
Ψ : = ( n r P ) ∧ ( n 2 p ) , (2) \Psi:=(n^r\mathbf P)\wedge(n^2\mathbf p), \tag*{(2)} Ψ := ( n r P ) ∧ ( n 2 p ) , (2) which has the following probabilistic interpretation
Ψ ≍ ( n r ) P ∧ ( n 2 ) p = E [ X n ( r ) ] ∧ E [ X n ( 2 ) ] . \Psi\asymp\binom{n}{r}\mathbb{P} \wedge\binom{n}{2}\mathbb{p} = \mathbb{E}[X_n^{(r)}] \wedge\mathbb{E}[X_n^{(2)}]. Ψ ≍ ( r n ) P ∧ ( 2 n ) p = E [ X n ( r ) ] ∧ E [ X n ( 2 ) ] . Furthermore, Ψ \Psi Ψ G ( n , p ) G(n,p) G ( n , p ) Φ \Phi Φ [9 ] ). However, contrary to the case of G ( n , p ) G(n,p) G ( n , p ) Ψ \Psi Ψ min 2 ≤ i ≤ r { E [ X n ( i ) ] } \min_{2\leq i\leq r}\{\mathbb{E}[X_n^{(i)}]\} min 2 ≤ i ≤ r { E [ X n ( i ) ]}
Below, we present our first result, i.e. the estimates of the variance of X n ( r ) X_n^{(r)} X n ( r )
Theorem 2.1. Let X n ( r ) X_n^{(r)} X n ( r ) r r r G ( n , m , p ) G(n,m,p) G ( n , m , p ) m p 3 ≤ c r mp^3\leq c_r m p 3 ≤ c r c r c_r c r r r r
Var [ X n ( r ) ] ≲ ( n r P ) 2 Ψ ( 1 − p ) + n 2 r − 1 ( P p ) 2 m p 3 ( 1 − p ) 2 . (3) \operatorname{Var}[X_n^{(r)}] \lesssim\frac{(n^r\mathbb{P})^2}{\Psi}(1-\mathbb{p})+n^{2r-1}\left(\frac{\mathbb{P}}{\mathbb{p}}\right)^2mp^3(1-\mathbb{p})^2. \tag*{(3)} Var [ X n ( r ) ] ≲ Ψ ( n r P ) 2 ( 1 − p ) + n 2 r − 1 ( p P ) 2 m p 3 ( 1 − p ) 2 . (3) The first term on the right-hand side of (3) is analogous to the estimate of the variance in G ( n , p ) G(n,p) G ( n , p ) G ( n , p ) G(n,p) G ( n , p ) G ( n , m , p ) G(n,m,p) G ( n , m , p )
Quantitative central limit theorems are usually described by means of the Kolmogorov distance or the Wasserstein one. For two random variables X X X Y Y Y d K ( X , Y ) d_K(X,Y) d K ( X , Y )
d K ( X , Y ) : = sup t ∈ R ∣ P ( X ≤ t ) − P ( Y ≤ t ) ∣ . d_K(X,Y):=\sup_{t\in\mathbb{R}}\left|\mathbb{P}(X\leq t)-\mathbb{P}(Y\leq t)\right|. d K ( X , Y ) := t ∈ R sup ∣ P ( X ≤ t ) − P ( Y ≤ t ) ∣ . while the Wasserstein distance d W ( X , Y ) d_W(X,Y) d W ( X , Y )
d W ( X , Y ) : = sup Lip ( h ) ≤ 1 ∣ E [ h ( X ) − h ( Y ) ] ∣ , d_W(X,Y):=\sup_{\operatorname{Lip}(h)\leq1}\left|\mathbb{E}[h(X)-h(Y)]\right|, d W ( X , Y ) := Lip ( h ) ≤ 1 sup ∣ E [ h ( X ) − h ( Y )] ∣ , where Lip ( h ) \operatorname{Lip}(h) Lip ( h ) h : R → R h:\mathbb{R}\to\mathbb{R} h : R → R
d K / W ( X , Y ) : = max { d K ( X , Y ) , d W ( X , Y ) } . d_{K/W}(X,Y):=\max\{d_K(X,Y),d_W(X,Y)\}. d K / W ( X , Y ) := max { d K ( X , Y ) , d W ( X , Y )} . The main result of this article is as follows.
Theorem 2.2. Assume m p 3 ≤ c r mp^3\leq c_r m p 3 ≤ c r c r c_r c r Theorem 2.1 . Then
X ~ n ( r ) → D N ⟺ ( 1 − p ) Ψ → ∞ , \widetilde{X}_n^{(r)} \xrightarrow{\mathcal{D}} \mathcal{N}\Longleftrightarrow(1-p)\Psi\to\infty, X n ( r ) D N ⟺ ( 1 − p ) Ψ → ∞ , where N ∼ N ( 0 , 1 ) \mathcal{N}\sim\mathcal{N}(0,1) N ∼ N ( 0 , 1 ) m 0 , n 0 ∈ N m_0,n_0\in\mathbb{N} m 0 , n 0 ∈ N C , C ′ > 0 C,C'>0 C , C ′ > 0 r r r n ≥ n 0 , m ≥ m 0 n\geq n_0,m\geq m_0 n ≥ n 0 , m ≥ m 0
d K / W ( X ~ n ( r ) , N ) ≤ C ( 1 ( 1 − p ) Ψ + 1 n + 1 n m p ) 1 / 2 , (4) d_{K/W}\left(\widetilde{X}_n^{(r)},\mathcal{N}\right)\leq C\left(\frac{1}{(1-p)\Psi}+\frac{1}{n}+\frac{1}{nmp}\right)^{1/2}, \tag*{(4)} d K / W ( X n ( r ) , N ) ≤ C ( ( 1 − p ) Ψ 1 + n 1 + nm p 1 ) 1/2 , (4) and
d K / W ( X ~ n ( r ) , N ) ≤ C ′ ( 1 ( 1 − p ) Ψ ) 1 / 4 . (5) d_{K/W}\left(\widetilde{X}_n^{(r)},\mathcal{N}\right)\leq C'\left(\frac{1}{(1-p)\Psi}\right)^{1/4}. \tag*{(5)} d K / W ( X n ( r ) , N ) ≤ C ′ ( ( 1 − p ) Ψ 1 ) 1/4 . (5) In the G ( n , p ) \mathcal{G}(n,\mathbf{p}) G ( n , p ) (4) takes the form ( ( 1 − p ) Ψ ) − 1 / 2 \left((1-\mathbf{p})\Psi\right)^{-1/2} ( ( 1 − p ) Ψ ) − 1/2 [17 , 19 ] . The complexity of the G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) n − 1 / 2 n^{-1/2} n − 1/2 n m p nmp nm p G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) [22 ] , where we considered only the case r = 2 r=2 r = 2 1 / m 1/m 1/ m 1 / ( n m p ) 1/(nmp) 1/ ( nm p ) (4) , which is worse whenever m p > 1 mp>1 m p > 1 [22 ] the calculations were simpler and yielded the minimum of these two terms, whereas we bounded it by 1 / m 1/m 1/ m
The organization of the remaining part of article is as follows. In § 3 , we begin by presenting Lemma 3.2, a result taken from [22 ] . It relates d K / W ( X ~ n ( r ) , N ) d_{K/W}(\widetilde{X}_{n}^{(r)},\mathcal{N}) d K / W ( X n ( r ) , N ) G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) Theorem 2.2 . § 4 is devoted to the development of techniques that yield sharp estimates for the probabilities of the graphs considered in Lemma 3.2 . In the subsequent two sections, we prove Theorem 2.1 and Theorem 2.2 , respectively. Both proofs rely on the results established in § 4 .
Preliminaries
Graphs G t , i , j G_{t,i,j} G t , i , j
In this section we define graphs G t , i , j G_{t,i,j} G t , i , j [22 ] . In general, for each t , i , j t,i,j t , i , j G t , i , j G_{t,i,j} G t , i , j K a b K_{ab} K ab a b ∈ Γ = { 11 , 12 , 21 , 22 } ab \in\Gamma=\{11,12,21,22\} ab ∈ Γ = { 11 , 12 , 21 , 22 } r r r
G t , i , j = ⋃ a b ∈ Γ K a b G_{t,i,j}=\bigcup_{ab\in\Gamma}K_{ab} G t , i , j = ab ∈ Γ ⋃ K ab where the cliques K a b K_{ab} K ab a b ∈ Γ ab \in\Gamma ab ∈ Γ t t t i i i j j j Figure 2 one may find the geometrical idea behind the definition, and in Figure 1 we present some particular cases.
Figure 1. Examples of G t , i , j G_{t,i,j} G t , i , j r = 5 r=5 r = 5 i = 2 i=2 i = 2 j = 4 j=4 j = 4
Figure 2. Symbolic depictions of graphs G t , i , j G_{t,i,j} G t , i , j
Definition 3.1. Let t = 1 , 2 , 3 t = 1, 2, 3 t = 1 , 2 , 3 a , b ∈ { 1 , 2 } a,b \in\{1,2\} a , b ∈ { 1 , 2 } G t , i , j G_{t,i,j} G t , i , j
(i) For $0 \leq i < j \leq r$, let
V ( G 1 , i , j ) = { v l , a : l ∈ { 1 , … , i } , a ∈ { 1 , 2 } } ∪ { w l : l ∈ { 1 , … , j − i } } ∪ { u l , a b : l ∈ { 1 , … , r − j } , a , b ∈ { 1 , 2 } } , \begin{aligned}
V(G_{1,i,j})={}&\{v_{l,a}:l\in\{1,\ldots,i\},a\in\{1,2\}\}\\
&\cup\{w_l:l\in\{1,\ldots,j-i\}\}\\
&\cup\{u_{l,ab}:l\in\{1,\ldots,r-j\},a,b\in\{1,2\}\},
\end{aligned} V ( G 1 , i , j ) = { v l , a : l ∈ { 1 , … , i } , a ∈ { 1 , 2 }} ∪ { w l : l ∈ { 1 , … , j − i }} ∪ { u l , ab : l ∈ { 1 , … , r − j } , a , b ∈ { 1 , 2 }} , and for each a , b ∈ { 1 , 2 } a,b \in\{1,2\} a , b ∈ { 1 , 2 }
K a b = G 1 , i , j [ { v l , a : l ∈ { 1 , … , i } } ∪ { w l : l ∈ { 1 , … , j − i } } ∪ { u l , a b : l ∈ { 1 , … , r − j } } ] K_{ab}=G_{1,i,j}[\{v_{l,a}:l\in\{1,\ldots,i\}\}\cup\{w_l:l\in\{1,\ldots,j-i\}\}\cup\{u_{l,ab}:l\in\{1,\ldots,r-j\}\}] K ab = G 1 , i , j [{ v l , a : l ∈ { 1 , … , i }} ∪ { w l : l ∈ { 1 , … , j − i }} ∪ { u l , ab : l ∈ { 1 , … , r − j }}] be a clique.
(ii) For $1 \leq i < j \leq r$, let
V ( G 2 , i , j ) = { v l , a : l ∈ { 1 , … , i } , a ∈ { 1 , 2 } } ∪ { w l , b : l ∈ { 1 , … , j − i } , b ∈ { 1 , 2 } } ∪ { u l , a b : l ∈ { 1 , … , r − j } , a , b ∈ { 1 , 2 } } . \begin{aligned}
V(G_{2,i,j})={}&\{v_{l,a}:l\in\{1,\ldots,i\},a\in\{1,2\}\}\\
&\cup\{w_{l,b}:l\in\{1,\ldots,j-i\},b\in\{1,2\}\}\\
&\cup\{u_{l,ab}:l\in\{1,\ldots,r-j\},a,b\in\{1,2\}\}.
\end{aligned} V ( G 2 , i , j ) = { v l , a : l ∈ { 1 , … , i } , a ∈ { 1 , 2 }} ∪ { w l , b : l ∈ { 1 , … , j − i } , b ∈ { 1 , 2 }} ∪ { u l , ab : l ∈ { 1 , … , r − j } , a , b ∈ { 1 , 2 }} . and for each a , b ∈ { 1 , 2 } a,b \in\{1,2\} a , b ∈ { 1 , 2 }
K a b = G 2 , i , j [ { v l , a : l ∈ { 1 , … , i } } ∪ { w l , b : l ∈ { 1 , … , j − i } } ∪ { u l , a b : l ∈ { 1 , … , r − j } } ] K_{ab}=G_{2,i,j}[\{v_{l,a}:l\in\{1,\ldots,i\}\}\cup\{w_{l,b}:l\in\{1,\ldots,j-i\}\}\cup\{u_{l,ab}:l\in\{1,\ldots,r-j\}\}] K ab = G 2 , i , j [{ v l , a : l ∈ { 1 , … , i }} ∪ { w l , b : l ∈ { 1 , … , j − i }} ∪ { u l , ab : l ∈ { 1 , … , r − j }}] be a clique.
(iii) For $1 \leq i < j \leq r$, let
V ( G 3 , i , j ) = { v l , a : l ∈ { 1 , … , i } , a ∈ { 1 , 2 } } ∪ { w l : l ∈ { 1 , … , j − i } } ∪ { w l , a : l ∈ { 1 , … , j − i } , a ∈ { 1 , 2 } } ∪ { u l , a b : l ∈ { 1 , … , r − j } , a , b ∈ { 1 , 2 } } . \begin{aligned}
V(G_{3,i,j})={}&\{v_{l,a}:l\in\{1,\ldots,i\},a\in\{1,2\}\}\\
&\cup\{w_l:l\in\{1,\ldots,j-i\}\}\\
&\cup\{w_{l,a}:l\in\{1,\ldots,j-i\},a\in\{1,2\}\}\\
&\cup\{u_{l,ab}:l\in\{1,\ldots,r-j\},a,b\in\{1,2\}\}.
\end{aligned} V ( G 3 , i , j ) = { v l , a : l ∈ { 1 , … , i } , a ∈ { 1 , 2 }} ∪ { w l : l ∈ { 1 , … , j − i }} ∪ { w l , a : l ∈ { 1 , … , j − i } , a ∈ { 1 , 2 }} ∪ { u l , ab : l ∈ { 1 , … , r − j } , a , b ∈ { 1 , 2 }} . and for each a ∈ { 1 , 2 } a \in\{1,2\} a ∈ { 1 , 2 }
K a 1 = G 3 , i , j [ { v l , a : l ∈ { 1 , … , i } } ∪ { w l : l ∈ { 1 , … , j − i } } ∪ { u l , a 1 : l ∈ { 1 , … , r − j } } ] K_{a1}=G_{3,i,j}[\{v_{l,a}:l\in\{1,\ldots,i\}\}\cup\{w_l:l\in\{1,\ldots,j-i\}\}\cup\{u_{l,a1}:l\in\{1,\ldots,r-j\}\}] K a 1 = G 3 , i , j [{ v l , a : l ∈ { 1 , … , i }} ∪ { w l : l ∈ { 1 , … , j − i }} ∪ { u l , a 1 : l ∈ { 1 , … , r − j }}] K a 2 = G 3 , i , j [ { v l , a : l ∈ { 1 , … , i } } ∪ { w l , a : l ∈ { 1 , … , j − i } } ∪ { u l , a 2 : l ∈ { 1 , … , r − j } } ] K_{a2}=G_{3,i,j}[\{v_{l,a}:l\in\{1,\ldots,i\}\}\cup\{w_{l,a}:l\in\{1,\ldots,j-i\}\}\cup\{u_{l,a2}:l\in\{1,\ldots,r-j\}\}] K a 2 = G 3 , i , j [{ v l , a : l ∈ { 1 , … , i }} ∪ { w l , a : l ∈ { 1 , … , j − i }} ∪ { u l , a 2 : l ∈ { 1 , … , r − j }}] be a clique.
Moreover no other edges except those of K a , b K_{a,b} K a , b a , b ∈ { 1 , 2 } a,b \in\{1,2\} a , b ∈ { 1 , 2 } G 1 , i , j G_{1,i,j} G 1 , i , j
In the sequel, we will omit indices i , j i,j i , j G t = G t , i , j G_t=G_{t,i,j} G t = G t , i , j i , j i,j i , j t = 1 t=1 t = 1 j = r j=r j = r K 11 = K 12 \mathbf{K}_{11}=\mathbf{K}_{12} K 11 = K 12 K 21 = K 22 \mathbf{K}_{21}=\mathbf{K}_{22} K 21 = K 22
Fact 3.1. G 1 , i , r G_{1,i,r} G 1 , i , r K r K_r K r r − i r-i r − i G 1 , 0 , r ≅ K r G_{1,0,r}\cong K_r G 1 , 0 , r ≅ K r
Given i , j i,j i , j t ∈ { 1 , 2 , 3 } t\in\{1,2,3\} t ∈ { 1 , 2 , 3 } I ⊆ Γ \mathbf{I}\subseteq\mathbf{\Gamma} I ⊆ Γ G t = G t , i , j G_t=G_{t,i,j} G t = G t , i , j
G t , I : = ⋃ a b ∈ I K a b , (6) G_{t,\mathbf{I}}:=\bigcup_{ab\in\mathbf{I}}K_{ab}, \tag*{(6)} G t , I := ab ∈ I ⋃ K ab , (6) i.e., G t , I G_{t,\mathbf{I}} G t , I G t G_t G t I \mathbf{I} I G t , ∅ G_{t,\varnothing} G t , ∅ π ( G t , ∅ ) = 1 \pi(G_{t,\varnothing})=1 π ( G t , ∅ ) = 1
The following lemma was proven in [22 ] .
Lemma 3.2 (Corollary 4.3 [22 ] ). Let t = 1 , 2 , 3 t=1,2,3 t = 1 , 2 , 3 G t = G t , i , j G_t=G_{t,i,j} G t = G t , i , j C ′ = C ′ ( r ) > 0 C'=C'(r)>0 C ′ = C ′ ( r ) > 0
d K / W ( X n ( r ) , N ) ≤ C ′ Var [ X n ( r ) ] ∑ t , i , j ( n ∣ V ( G t ) ∣ ∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) ) . d_{K/W}\left(X_n^{(r)},\mathcal{N}\right)\leq\frac{C'}{\operatorname{Var}[X_n^{(r)}]}\sum_{t,i,j}\left(n^{|V(G_t)|}\sum_{\mathbf{I}\subseteq\mathbf{\Gamma}}(-1)^{|\mathbf{I}|}\mathbf{P}^{|\mathbf{\Gamma}|-|\mathbf{I}|}\pi(G_{t,\mathbf{I}})\right). d K / W ( X n ( r ) , N ) ≤ Var [ X n ( r ) ] C ′ t , i , j ∑ ( n ∣ V ( G t ) ∣ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) ) . In the remainder of the paper, we focus on obtaining estimates for the quantity appearing on the right-hand side of the equation in the lemma.
Subgraph probabilities
In this section, we present general results concerning probabilities of subgraph occurrence in G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) Var [ X n ( r ) ] \operatorname{Var}[X_n^{(r)}] Var [ X n ( r ) ] G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p )
In the following discussion, we will determine the probabilities of graphs that can be represented as unions of graphs. For such graphs, we introduce new notation.
Definition 4.1. For two graphs H 1 H_1 H 1 H 2 H_2 H 2 H 1 ′ H'_1 H 1 ′ H 2 ′ H'_2 H 2 ′ H 1 ′ ∩ H 2 ′ H'_1 \cap H'_2 H 1 ′ ∩ H 2 ′ i i i i ≥ 0 i \ge0 i ≥ 0 H 1 ⊕ i H 2 = H 1 ′ ∪ H 2 ′ H_1 \oplus_i H_2 = H'_1 \cup H'_2 H 1 ⊕ i H 2 = H 1 ′ ∪ H 2 ′ H 1 H_1 H 1 H 2 H_2 H 2 i i i H 1 ⊕ i , j H 2 H_1 \oplus_{i,j} H_2 H 1 ⊕ i , j H 2 i , j ≥ 1 i,j \ge1 i , j ≥ 1 H 1 H_1 H 1 H 2 H_2 H 2 i i i j j j Figure 3 ).
Figure 3. Examples of K 3 ⊕ 2 K 3 ⊕ 0 K 2 K_3 \oplus_2 K_3 \oplus_0 K_2 K 3 ⊕ 2 K 3 ⊕ 0 K 2 ( K 3 ⊕ 1 K 3 ) ⊕ 1 , 1 ( K 3 ⊕ 1 K 3 ) ⊕ 0 N 1 (K_3 \oplus_1 K_3) \oplus_{1,1} (K_3 \oplus_1 K_3) \oplus_0 N_1 ( K 3 ⊕ 1 K 3 ) ⊕ 1 , 1 ( K 3 ⊕ 1 K 3 ) ⊕ 0 N 1
In Definition 4.1, formally we consider a new vertex set and isomorphic copies of H 1 H_1 H 1 H 2 H_2 H 2 H 1 H_1 H 1 H 2 H_2 H 2 H 1 ⊕ i H 2 H_1 \oplus_i H_2 H 1 ⊕ i H 2 H 1 ⊕ i , j H 2 H_1 \oplus_{i,j} H_2 H 1 ⊕ i , j H 2
Clique covers
In this section, we define the clique covers introduced in [10 ] with some modifications proposed in [23 ] . They play a crucial role in determining the probabilities of subgraphs in G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p )
Definition 4.2. For a given graph H H H P ( H ) \mathcal{P}(H) P ( H ) V ( H ) V(H) V ( H ) H H H C ∈ P ( H ) C \in\mathcal{P}(H) C ∈ P ( H ) t t t t t t C ⊆ P ( H ) \mathcal{C} \subseteq\mathcal{P}(H) C ⊆ P ( H ) clique cover of H H H e ∈ E ( H ) e \in E(H) e ∈ E ( H ) C ∈ C C \in\mathcal{C} C ∈ C H H H C ( H ) \mathcal{C}(H) C ( H )
For a graph H H H V ( H ) ⊆ V V(H) \subseteq\mathcal{V} V ( H ) ⊆ V a ∈ A a \in\mathcal{A} a ∈ A builds C ⊆ V ( H ) C \subseteq V(H) C ⊆ V ( H ) C C C H H H a a a C = { v ∈ V ( H ) : v chose a } C = \{v \in V(H) : v \text{ chose } a\} C = { v ∈ V ( H ) : v chose a }
p C : = p ∣ C ∣ ( 1 − p ) ∣ V ( H ) ∣ − ∣ C ∣ . (7) p_C := p^{|C|}(1-p)^{|V(H)|-|C|}. \tag*{(7)} p C := p ∣ C ∣ ( 1 − p ) ∣ V ( H ) ∣ − ∣ C ∣ . (7) Then, H H H given by a clique cover C \mathcal{C} C G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) C ∈ C C \in\mathcal{C} C ∈ C A \mathcal{A} A A \mathcal{A} A C ∈ P ( H ) ∖ C C \in\mathcal{P}(H) \setminus\mathcal{C} C ∈ P ( H ) ∖ C
By π ( H , C ) \pi(H,\mathcal{C}) π ( H , C ) H H H C \mathcal{C} C G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p ) C ⊆ C ( H ) \mathcal{C} \subseteq\mathcal{C}(H) C ⊆ C ( H ) π ( H , C ) \pi(H,\mathcal{C}) π ( H , C ) H H H C \mathcal{C} C
π ( H , C ) = ∑ C ∈ C π ( H , C ) . (8) \pi(H,\mathcal{C}) = \sum_{\mathcal{C}\in\mathcal{C}} \pi(H,\mathcal{C}). \tag*{(8)} π ( H , C ) = C ∈ C ∑ π ( H , C ) . (8) In particular, the probability that H H H G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p )
π ( H ) : = P ( H ⊆ G ( n , m , p ) ) = π ( H , C ( H ) ) = ∑ C ∈ C ( H ) π ( H , C ) . (9) \pi(H) := \mathbb{P}\left(H \subseteq\mathcal{G}(n,m,p)\right) = \pi(H,\mathcal{C}(H)) = \sum_{\mathcal{C}\in\mathcal{C}(H)} \pi(H,\mathcal{C}). \tag*{(9)} π ( H ) := P ( H ⊆ G ( n , m , p ) ) = π ( H , C ( H )) = C ∈ C ( H ) ∑ π ( H , C ) . (9) Next, for a clique cover C + ⊆ P ( H ) \mathcal{C}_{+} \subseteq\mathcal{P}(H) C + ⊆ P ( H ) C − ⊆ P ( H ) ∖ C + \mathcal{C}_{-} \subseteq\mathcal{P}(H) \setminus\mathcal{C}_{+} C − ⊆ P ( H ) ∖ C + C ( H , C + , C − ) \mathcal{C}(H,\mathcal{C}_{+},\mathcal{C}_{-}) C ( H , C + , C − ) C \mathcal{C} C C + \mathcal{C}_{+} C + C − \mathcal{C}_{-} C − C + ⊆ C ⊆ P ( H ) ∖ C − \mathcal{C}_{+} \subseteq\mathcal{C} \subseteq\mathcal{P}(H) \setminus\mathcal{C}_{-} C + ⊆ C ⊆ P ( H ) ∖ C − C − \mathcal{C}_{-} C − forbidden . Additionally, we will be using the shorthand notation
π ( H , C + , C − ) : = π ( H , C ( H , C + , C − ) ) = ∑ C ∈ C ( H , C + , C − ) π ( H , C ) . \pi(H,\mathcal{C}_{+},\mathcal{C}_{-}) := \pi(H,\mathcal{C}(H,\mathcal{C}_{+},\mathcal{C}_{-})) = \sum_{\mathcal{C}\in\mathcal{C}(H,\mathcal{C}_{+},\mathcal{C}_{-})} \pi(H,\mathcal{C}). π ( H , C + , C − ) := π ( H , C ( H , C + , C − )) = C ∈ C ( H , C + , C − ) ∑ π ( H , C ) . In particular, we have the following identity π ( H , C ) = π ( H , C , P ( H ) ∖ C ) \pi(H,\mathcal{C}) = \pi(H,\mathcal{C},\mathcal{P}(H) \setminus\mathcal{C}) π ( H , C ) = π ( H , C , P ( H ) ∖ C )
The following asymptotics were shown in [23 ] . See also [10 ] , [25 ] , [6 ] for some previous, less general results.
Lemma 4.3 ([23 ] ). Let H H H P ( H ) \mathcal{P}(H) P ( H ) V ( H ) V(H) V ( H ) H H H C + \mathcal{C}_{+} C + C − ⊆ P ( H ) ∖ C + \mathcal{C}_{-} \subseteq\mathcal{P}(H) \setminus\mathcal{C}_{+} C − ⊆ P ( H ) ∖ C + m → ∞ m \to\infty m → ∞ p p p
π ( H , C + , C − ) = ( 1 + O ( m − δ ∣ V ( H ) ∣ ) ) ( 1 − ∑ C ∈ C − p C ) m ∏ C ∈ C + ( 1 − e − m p C ) , \pi(H,\mathcal{C}_{+},\mathcal{C}_{-}) = \left(1+O\left(m^{-\delta_{|V(H)|}}\right)\right)\left(1-\sum_{\mathcal{C}\in\mathcal{C}_{-}}p_{\mathcal{C}}\right)^m \prod_{\mathcal{C}\in\mathcal{C}_{+}}\left(1-e^{-mp_{\mathcal{C}}}\right), π ( H , C + , C − ) = ( 1 + O ( m − δ ∣ V ( H ) ∣ ) ) 1 − C ∈ C − ∑ p C m C ∈ C + ∏ ( 1 − e − m p C ) , where p C p_{\mathcal{C}} p C (7) , and δ k = 1 / ( k 2 k + 1 ) \delta_k = 1/(k2^{k+1}) δ k = 1/ ( k 2 k + 1 )
Moreover, for m p 2 ≤ 1 mp^2 \leq1 m p 2 ≤ 1
π ( H , C + , C − ) ≍ π ( H , C + ) ≍ ∏ C ∈ C + m p ∣ C ∣ = m ∣ C + ∣ p ∑ C ∈ C + ∣ C ∣ , (10) \pi(H,\mathcal{C}_{+},\mathcal{C}_{-}) \asymp\pi(H,\mathcal{C}_{+}) \asymp\prod_{\mathcal{C}\in\mathcal{C}_{+}} mp^{|\mathcal{C}|} = m^{|\mathcal{C}_{+}|}p^{\sum_{\mathcal{C}\in\mathcal{C}_{+}}|\mathcal{C}|}, \tag*{(10)} π ( H , C + , C − ) ≍ π ( H , C + ) ≍ C ∈ C + ∏ m p ∣ C ∣ = m ∣ C + ∣ p ∑ C ∈ C + ∣ C ∣ , (10) and
π ( H , E ( H ) ) = ( 1 + O ( m − δ ∣ V ( H ) ∣ ) ) p ∣ E ( H ) ∣ , (11) \pi(H,E(H)) = \left(1+O\left(m^{-\delta_{|V(H)|}}\right)\right)p^{|E(H)|}, \tag*{(11)} π ( H , E ( H )) = ( 1 + O ( m − δ ∣ V ( H ) ∣ ) ) p ∣ E ( H ) ∣ , (11) where δ k = 1 / ( k 2 k + 1 ) \delta_k = 1/(k2^{k+1}) δ k = 1/ ( k 2 k + 1 )
Furthermore, for m p 2 ≥ 1 mp^2 \geq1 m p 2 ≥ 1 m p 3 ≤ 1 mp^3 \leq1 m p 3 ≤ 1
π ( H , C + , C − ) ≍ e − ∣ C + ( 2 ) ∣ m p 2 ∏ C ∈ C + ∖ C + ( 2 ) m p ∣ C ∣ , (12) \pi(H,\mathcal{C}_{+},\mathcal{C}_{-}) \asymp e^{-|\mathcal{C}_{+}^{(2)}|mp^2}\prod_{\mathcal{C}\in\mathcal{C}_{+}\setminus\mathcal{C}_{+}^{(2)}}mp^{|\mathcal{C}|}, \tag*{(12)} π ( H , C + , C − ) ≍ e − ∣ C + ( 2 ) ∣ m p 2 C ∈ C + ∖ C + ( 2 ) ∏ m p ∣ C ∣ , (12) where C + ( 2 ) \mathcal{C}_{+}^{(2)} C + ( 2 ) C − ( 2 ) \mathcal{C}_{-}^{(2)} C − ( 2 ) C + \mathcal{C}_{+} C + C − \mathcal{C}_{-} C −
We close this section with another result from [23 ] . For two families of sets C \mathcal{C} C C ′ \mathcal{C}' C ′
C ≐ C ′ , (13) \mathcal{C} \doteq\mathcal{C}', \tag*{(13)} C ≐ C ′ , (13) if there is a bijection f : C → C ′ f:\mathcal{C}\to\mathcal{C}' f : C → C ′ ∣ f ( C ) ∣ = ∣ C ∣ |f(\mathcal{C})|=|\mathcal{C}| ∣ f ( C ) ∣ = ∣ C ∣ C ∈ C \mathcal{C}\in\mathcal{C} C ∈ C
Lemma 4.4 (Proposition 2.3 [23 ] ). Let H 1 H_1 H 1 H 2 H_2 H 2 i = 1 , 2 i=1,2 i = 1 , 2 C + ( i ) \mathcal{C}_{+}^{(i)} C + ( i ) H i H_i H i C − ( i ) ⊆ P ( H i ) ∖ C + ( i ) \mathcal{C}_{-}^{(i)} \subseteq\mathcal{P}(H_i) \setminus\mathcal{C}_{+}^{(i)} C − ( i ) ⊆ P ( H i ) ∖ C + ( i ) C + ( 1 ) ≐ C + ( 2 ) \mathcal{C}_{+}^{(1)} \doteq\mathcal{C}_{+}^{(2)} C + ( 1 ) ≐ C + ( 2 ) C − ( 1 ) ≐ C − ( 2 ) \mathcal{C}_{-}^{(1)} \doteq\mathcal{C}_{-}^{(2)} C − ( 1 ) ≐ C − ( 2 )
π ( H 1 , C + ( 1 ) , C − ( 1 ) ) = π ( H 2 , C + ( 2 ) , C − ( 2 ) ) . \pi(H_1,\mathcal{C}_{+}^{(1)},\mathcal{C}_{-}^{(1)}) = \pi(H_2,\mathcal{C}_{+}^{(2)},\mathcal{C}_{-}^{(2)}). π ( H 1 , C + ( 1 ) , C − ( 1 ) ) = π ( H 2 , C + ( 2 ) , C − ( 2 ) ) . Basic subgraph probabilities
This section collects various general results concerning subgraph probabilities.
Fact 4.1. For any graphs H 1 H_1 H 1 H 2 H_2 H 2
π ( H 1 ∪ H 2 ) ≥ π ( H 1 ) π ( H 2 ) . \pi(H_1 \cup H_2) \ge\pi(H_1)\pi(H_2). π ( H 1 ∪ H 2 ) ≥ π ( H 1 ) π ( H 2 ) . Proof. The indicators 1 { H 1 ⊆ G ( n , m , p ) } \mathbf{1}_{\{H_1\subseteq G(n,m,p)\}} 1 { H 1 ⊆ G ( n , m , p )} 1 { H 2 ⊆ G ( n , m , p ) } \mathbf{1}_{\{H_2\subseteq G(n,m,p)\}} 1 { H 2 ⊆ G ( n , m , p )} [9 ] ), we have
E [ 1 { H 1 ⊆ G ( n , m , p ) } 1 { H 2 ⊆ G ( n , m , p ) } ] ≥ E [ 1 { H 1 ⊆ G ( n , m , p ) } ] E [ 1 { H 2 ⊆ G ( n , m , p ) } ] , \mathbb{E}[\mathbf{1}_{\{H_1\subseteq G(n,m,p)\}}\mathbf{1}_{\{H_2\subseteq G(n,m,p)\}}] \ge\mathbb{E}[\mathbf{1}_{\{H_1\subseteq G(n,m,p)\}}]\mathbb{E}[\mathbf{1}_{\{H_2\subseteq G(n,m,p)\}}], E [ 1 { H 1 ⊆ G ( n , m , p )} 1 { H 2 ⊆ G ( n , m , p )} ] ≥ E [ 1 { H 1 ⊆ G ( n , m , p )} ] E [ 1 { H 2 ⊆ G ( n , m , p )} ] , which is equivalent to the assertion.
The following lemma was proved in [10 ] under an additional assumption on m m m [10 ] can be adapted to yield the version stated below, see also [23 ] . The lemma can also be obtained using the methods developed in this paper.
Lemma 4.5. For K ≅ K r \mathbf{K} \cong K_r K ≅ K r
P = π ( K ) ≍ { π ( K , { V ( K ) } ) ≍ m p r for p ≤ m − r + 1 2 r , π ( K , E ( K ) ) ≍ p ( r 2 ) for p ≥ m − r + 1 2 r . \mathbf{P}=\pi(\mathbf{K}) \asymp
\begin{cases}
\pi(\mathbf{K},\{V(\mathbf{K})\}) \asymp mp^r & \text{for } p \le m^{-\frac{r+1}{2r}},\\
\pi(\mathbf{K},E(\mathbf{K})) \asymp p^{\binom{r}{2}} & \text{for } p \ge m^{-\frac{r+1}{2r}}.
\end{cases} P = π ( K ) ≍ { π ( K , { V ( K )}) ≍ m p r π ( K , E ( K )) ≍ p ( 2 r ) for p ≤ m − 2 r r + 1 , for p ≥ m − 2 r r + 1 . Let us note that due to the estimate p ≍ m p 2 p \asymp mp^2 p ≍ m p 2 m p 2 < 1 mp^2<1 m p 2 < 1 P = π ( K ) ≍ ( m p r + p ( r 2 ) ) \mathbf{P}=\pi(\mathbf{K})\asymp(mp^r+p^{\binom{r}{2}}) P = π ( K ) ≍ ( m p r + p ( 2 r ) )
It will be convenient to define
Ψ i = Ψ i ( n , m , p , r ) = { n i m p i ≍ n i π ( K i , { V ( K i ) } ) for p ≤ m − r + 1 2 r , n i p ( i 2 ) ≍ n i π ( K i , E ( K i ) ) for p ≥ m − r + 1 2 r . (14) \Psi_i=\Psi_i(n,m,p,r)=
\begin{cases}
n^i mp^i \asymp n^i\pi(K_i,\{V(K_i)\}) & \text{for } p \le m^{-\frac{r+1}{2r}},\\
n^i p^{\binom{i}{2}} \asymp n^i\pi(K_i,E(K_i)) & \text{for } p \ge m^{-\frac{r+1}{2r}}.
\end{cases} \tag*{(14)} Ψ i = Ψ i ( n , m , p , r ) = { n i m p i ≍ n i π ( K i , { V ( K i )}) n i p ( 2 i ) ≍ n i π ( K i , E ( K i )) for p ≤ m − 2 r r + 1 , for p ≥ m − 2 r r + 1 . (14) The following conclusion concerning Ψ \Psi Ψ
min 2 ≤ i ≤ r Ψ i ≍ Ψ , (15) \min_{2\le i\le r}\Psi_i\asymp\Psi, \tag*{(15)} 2 ≤ i ≤ r min Ψ i ≍ Ψ , (15) which is a consequence of the fact that for s ∈ [ 0 , 1 ] s\in[0,1] s ∈ [ 0 , 1 ] min 2 ≤ i ≤ r { n i s ( 2 ) ( i ) } = ( n 2 s ) ∧ ( n r s ( 2 ) ( r ) ) \min_{2\le i\le r}\{n^i s_{(2)}^{(i)}\}=(n^2s)\wedge(n^r s_{(2)}^{(r)}) min 2 ≤ i ≤ r { n i s ( 2 ) ( i ) } = ( n 2 s ) ∧ ( n r s ( 2 ) ( r ) ) [9 ] or Proposition 4.7 in [17 ] . Let us note that Lemma 4.5 implies that Ψ i \Psi_i Ψ i E [ X n ( i ) ] \mathbb{E}[X_n^{(i)}] E [ X n ( i ) ] min 2 ≤ i ≤ r E [ X n ( i ) ] \min_{2\le i\le r}\mathbb{E}[X_n^{(i)}] min 2 ≤ i ≤ r E [ X n ( i ) ] Ψ \Psi Ψ G ( n , p ) \mathcal{G}(n,p) G ( n , p )
The next result will be mostly useful in analysing subgraph probabilities in the case when m p 2 mp^2 m p 2 p ≍ 1 p\asymp1 p ≍ 1
Fact 4.2. For any non-empty graph H H H
p ∣ E ( H ) ∣ ≤ π ( H ) ≤ p , p^{|E(H)|}\le\pi(H)\le p, p ∣ E ( H ) ∣ ≤ π ( H ) ≤ p , and
1 − π ( H ) ≍ 1 − p . 1-\pi(H)\asymp1-p. 1 − π ( H ) ≍ 1 − p . Proof. The inequality π ( H ) ≤ p \pi(H) \le p π ( H ) ≤ p { H ⊆ G ( n , m , p ) } ⊆ { e ∈ G ( n , m , p ) } \{H \subseteq\mathcal{G}(n,m,p)\} \subseteq\{e \in\mathcal{G}(n,m,p)\} { H ⊆ G ( n , m , p )} ⊆ { e ∈ G ( n , m , p )} e ∈ E ( H ) e \in E(H) e ∈ E ( H ) p ∣ E ( H ) ∣ ≤ π ( H ) \mathbf{p}^{|E(H)|} \le\pi(H) p ∣ E ( H ) ∣ ≤ π ( H )
1 − p ≤ 1 − π ( H ) ≤ 1 − p ∣ E ( H ) ∣ ≤ ∣ E ( H ) ∣ ( 1 − p ) , 1-p \le1-\pi(H) \le1-\mathbf{p}^{|E(H)|} \le|E(H)|(1-p), 1 − p ≤ 1 − π ( H ) ≤ 1 − p ∣ E ( H ) ∣ ≤ ∣ E ( H ) ∣ ( 1 − p ) , which completes the proof.
Comparing clique covers
In this section, we develop techniques for identifying which clique covers of a given graph are most probable, i.e., those that contribute most to the sum (9) . In particular, these methods enable the determination of the variance and covariance of the indicators of a given subgraph’s existence in G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p )
For C ∈ C ( H ) \mathcal{C} \in\mathcal{C}(H) C ∈ C ( H ) V ′ ⊆ V ( H ) V' \subseteq V(H) V ′ ⊆ V ( H ) H ′ = H [ V ′ ] H'=H[V'] H ′ = H [ V ′ ]
C [ V ′ ] = { C ∩ V ′ : ∃ e ∈ E ( H ′ ) e ⊆ C } . \mathcal{C}[V'] = \{C \cap V' : \exists e \in E(H')\, e \subseteq C\}. C [ V ′ ] = { C ∩ V ′ : ∃ e ∈ E ( H ′ ) e ⊆ C } . Furthermore, for C ′ ⊆ C \mathcal{C}' \subseteq\mathcal{C} C ′ ⊆ C C ′ ′ ⊆ P ( V ( H ) ) \mathcal{C}'' \subseteq\mathcal{P}(V(H)) C ′′ ⊆ P ( V ( H )) C [ C ′ ↦ C ′ ′ ] \mathcal{C}[\mathcal{C}' \mapsto\mathcal{C}''] C [ C ′ ↦ C ′′ ] C \mathcal{C} C
C [ C ′ ↦ C ′ ′ ] = ( C ∖ C ′ ) ∪ C ′ ′ . \mathcal{C}[\mathcal{C}' \mapsto\mathcal{C}''] = (\mathcal{C} \setminus\mathcal{C}') \cup\mathcal{C}''. C [ C ′ ↦ C ′′ ] = ( C ∖ C ′ ) ∪ C ′′ . In what follows, C ′ \mathcal{C}' C ′ C ′ ′ \mathcal{C}'' C ′′ C [ C ′ ↦ C ′ ′ ] \mathcal{C}[\mathcal{C}' \mapsto\mathcal{C}''] C [ C ′ ↦ C ′′ ] H H H
Definition 4.6. Let H H H C ∈ C ( H ) \mathcal{C} \in\mathcal{C}(H) C ∈ C ( H ) simple clique cover of H H H
P1 ∀ C 1 , C 2 ∈ C ∣ C 1 ∩ C 2 ∣ ≤ 1 \forall C_1,C_2 \in\mathcal{C}\ |C_1 \cap C_2| \le1 ∀ C 1 , C 2 ∈ C ∣ C 1 ∩ C 2 ∣ ≤ 1
P2 ∀ C ∈ C H [ C ] \forall C \in\mathcal{C}\ H[C] ∀ C ∈ C H [ C ]
Let us note that P1 implies that each edge is covered by at most one set from C \mathcal{C} C
Lemma 4.7. Let m p 2 ≤ 1 mp^2 \le1 m p 2 ≤ 1 H H H C ∈ C ( H ) \mathcal{C} \in\mathcal{C}(H) C ∈ C ( H )
(i) There exists a simple C ′ ∈ C ( H ) \mathcal{C}' \in\mathcal{C}(H) C ′ ∈ C ( H )
π ( H , C ) ≲ π ( H , C ′ ) . \pi(H,\mathcal{C}) \lesssim\pi(H,\mathcal{C}'). π ( H , C ) ≲ π ( H , C ′ ) . (ii) For every clique cover with no P1 there exists C ′ ∈ C ( H ) \mathcal{C}' \in\mathcal{C}(H) C ′ ∈ C ( H )
π ( H , C ) ≲ π ( H , C ′ ) , ∀ C ∈ C ∃ C ′ ∈ C ′ C ⊆ C ′ , ∣ C ′ ∣ ≤ ∣ C ∣ − 1. (16) \pi(H,\mathcal{C}) \lesssim\pi(H,\mathcal{C}'), \qquad\forall C \in\mathcal{C}\ \exists C' \in\mathcal{C}'\ C \subseteq C', \qquad|\mathcal{C}'| \le|\mathcal{C}|-1. \tag*{(16)} π ( H , C ) ≲ π ( H , C ′ ) , ∀ C ∈ C ∃ C ′ ∈ C ′ C ⊆ C ′ , ∣ C ′ ∣ ≤ ∣ C ∣ − 1. (16) Additionally, if C \mathcal{C} C C ′ \mathcal{C}' C ′
Proof of Lemma 4.7. We start with the proof of (ii). Assume that C \mathcal{C} C P1 . Then, there are two sets C ′ , C ′ ′ ∈ C C', C'' \in\mathcal{C} C ′ , C ′′ ∈ C (10) , we get
π ( H , C ) π ( H , C [ { C ′ , C ′ ′ } ↦ { C ′ ∪ C ′ ′ } ] ) ≍ m p ∣ C ′ ∣ m p ∣ C ′ ′ ∣ m p ∣ C ′ ∪ C ′ ′ ∣ = m p ∣ C ′ ∩ C ′ ′ ∣ ≤ m p 2 ≤ 1. \frac{\pi(H,\mathcal{C})}{\pi(H,\mathcal{C}[\{C',C''\}\mapsto\{C'\cup C''\}])} \asymp\frac{mp^{|C'|}mp^{|C''|}}{mp^{|C'\cup C''|}}=mp^{|C'\cap C''|}\le mp^2\le1. π ( H , C [{ C ′ , C ′′ } ↦ { C ′ ∪ C ′′ }]) π ( H , C ) ≍ m p ∣ C ′ ∪ C ′′ ∣ m p ∣ C ′ ∣ m p ∣ C ′′ ∣ = m p ∣ C ′ ∩ C ′′ ∣ ≤ m p 2 ≤ 1. Applying this procedure inductively, we arrive at a clique cover C \mathcal{C} C P1 and satisfying (16) . Moreover, if C \mathcal{C} C P2 , then the graphs H [ C ′ ] H[C'] H [ C ′ ] H [ C ′ ′ ] H[C''] H [ C ′′ ] H [ C ′ ∪ C ′ ′ ] H[C'\cup C''] H [ C ′ ∪ C ′′ ] C ′ \mathcal{C}' C ′ P2 .
In view of (ii), in order to prove (i) it is enough to show existence of C ′ \mathcal{C}' C ′ P2 . If C \mathcal{C} C P2 , there exists C ∈ C C\in\mathcal{C} C ∈ C H [ C ] H[C] H [ C ] C 1 , … , C k C_1,\ldots,C_k C 1 , … , C k C 1 ∪ C 2 ∪ … ∪ C k ⊆ C C_1\cup C_2\cup\ldots\cup C_k\subseteq C C 1 ∪ C 2 ∪ … ∪ C k ⊆ C ∣ C i ∣ ≥ 2 |C_i|\ge2 ∣ C i ∣ ≥ 2 i = 1 , … , k i=1,\ldots,k i = 1 , … , k C 1 ∪ C 2 ∪ … ∪ C k C_1\cup C_2\cup\ldots\cup C_k C 1 ∪ C 2 ∪ … ∪ C k H [ C ] H[C] H [ C ] H [ C i ] H[C_i] H [ C i ] i = 1 , … , k i=1,\ldots,k i = 1 , … , k H [ C ] H[C] H [ C ] k ≥ 1 k\ge1 k ≥ 1 (10) , for C ′ = C [ { C } ↦ { C 1 , … , C k } ] \mathcal{C}'=\mathcal{C}[\{C\}\mapsto\{C_1,\ldots,C_k\}] C ′ = C [{ C } ↦ { C 1 , … , C k }]
π ( H , C ) π ( H , C ′ ) ≍ m p ∣ C ∣ m k p ∣ C 1 ∣ + … + ∣ C k ∣ = p ∣ C ∣ − ∣ C 1 ∣ − … − ∣ C k ∣ m k − 1 ≤ 1. \frac{\pi(H,\mathcal{C})}{\pi(H,\mathcal{C}')} \asymp\frac{mp^{|C|}}{m^kp^{|C_1|+\ldots+|C_k|}}=\frac{p^{|C|-|C_1|-\ldots-|C_k|}}{m^{k-1}}\le1. π ( H , C ′ ) π ( H , C ) ≍ m k p ∣ C 1 ∣ + … + ∣ C k ∣ m p ∣ C ∣ = m k − 1 p ∣ C ∣ − ∣ C 1 ∣ − … − ∣ C k ∣ ≤ 1. Since H [ C 1 ] , … , H [ C k ] H[C_1],\ldots,H[C_k] H [ C 1 ] , … , H [ C k ] P2 . □ \square □
The remainder of the section is primarily devoted to the proof of the lemma below. For m p 2 ≤ 1 mp^2\le1 m p 2 ≤ 1 G t G_t G t K a b \mathbf{K}_{ab} K ab
Lemma 4.9. Assume m p 2 ≤ 1 mp^2\le1 m p 2 ≤ 1 j − i ≥ 2 j-i\ge2 j − i ≥ 2 i ≥ 0 i\ge0 i ≥ 0 t = 1 t=1 t = 1 i ≥ 2 i\ge2 i ≥ 2 t = 2 , 3 t=2,3 t = 2 , 3 t t t I \mathbf{I} I ∣ I ∣ ≥ 2 |\mathbf{I}|\ge2 ∣ I ∣ ≥ 2 G t , I G_{t,\mathbf{I}} G t , I C ∈ C ( G t , I ) \mathcal{C}\in\mathcal{C}(G_{t,\mathbf{I}}) C ∈ C ( G t , I ) a b ∈ I ab\in\mathbf{I} ab ∈ I
n ∣ V ( G t , I ) ∣ π ( G t , I , C ) ≲ n r P ⋅ n ∣ V ( G t , I ∖ { a b } ) ∣ π ( G t , I ∖ { a b } ) Ψ , n^{|V(G_{t,\mathbf{I}})|}\pi(G_{t,\mathbf{I}},\mathcal{C})\lesssim\frac{n^{r}\mathbf{P}\cdot n^{|V(G_{t,\mathbf{I}\setminus\{ab\}})|}\pi(G_{t,\mathbf{I}\setminus\{ab\}})}{\Psi}, n ∣ V ( G t , I ) ∣ π ( G t , I , C ) ≲ Ψ n r P ⋅ n ∣ V ( G t , I ∖ { ab } ) ∣ π ( G t , I ∖ { ab } ) , and G t , I ∖ { a b } G_{t,\mathbf{I}\setminus\{ab\}} G t , I ∖ { ab }
Remark 4.10. In the case t = 1 t=1 t = 1 r = j r=j r = j ∣ I ∣ ≥ 3 |\mathbf{I}|\ge3 ∣ I ∣ ≥ 3 I ∈ { { 11 , 12 } , { 21 , 22 } } \mathbf{I}\in\{\{11,12\},\{21,22\}\} I ∈ {{ 11 , 12 } , { 21 , 22 }} a b ab ab G t , I ∖ { a b } = G t , I G_{t,\mathbf{I}\setminus\{ab\}}=G_{t,\mathbf{I}} G t , I ∖ { ab } = G t , I Ψ ≤ n r P \Psi\le n^r\mathbf{P} Ψ ≤ n r P i = 0 i=0 i = 0 I \mathbf{I} I ∣ I ∣ = 2 |\mathbf{I}|=2 ∣ I ∣ = 2
We begin with a lemma that provides simple estimates of clique cover probabilities when coverings by 2-sets are more likely, and when coverings by larger sets are more probable.
Lemma 4.11. Let m p 2 ≤ 1 mp^2\le1 m p 2 ≤ 1 H H H C ∈ C ( H ) \mathcal{C}\in\mathcal{C}(H) C ∈ C ( H ) C ∈ C C\in\mathcal{C} C ∈ C ∣ C ∣ ≥ 3 |C|\ge3 ∣ C ∣ ≥ 3
(i) If p ≤ m ∣ C ∣ + 1 2 ∣ C ∣ p \le m^{\frac{|C|+1}{2|C|}} p ≤ m 2∣ C ∣ ∣ C ∣ + 1 H [ C ] H[C] H [ C ]
π ( H , C [ { C } ↦ E ( H [ C ] ) ] ) ≲ π ( H , C ) . \pi(H,\mathcal{C}[\{C\}\mapsto E(H[C])]) \lesssim\pi(H,\mathcal{C}). π ( H , C [{ C } ↦ E ( H [ C ])]) ≲ π ( H , C ) . (ii) If p ≥ m ∣ C ∣ + 1 2 ∣ C ∣ p \ge m^{\frac{|C|+1}{2|C|}} p ≥ m 2∣ C ∣ ∣ C ∣ + 1
π ( H , C ) ≲ π ( H , C [ { C } ↦ E ( H [ C ] ) ] ) . \pi(H,\mathcal{C}) \lesssim\pi(H,\mathcal{C}[\{C\}\mapsto E(H[C])]). π ( H , C ) ≲ π ( H , C [{ C } ↦ E ( H [ C ])]) . (iii) If A m − h + 1 2 h ≤ p ≤ m − 1 / 2 Am^{-\frac{h+1}{2h}} \le p \le m^{-1/2} A m − 2 h h + 1 ≤ p ≤ m − 1/2 h = ∣ V ( H ) ∣ h=|V(H)| h = ∣ V ( H ) ∣ A ≥ 1 A\ge1 A ≥ 1
π ( H ) = ( 1 + O ( A − 1 ) ) π ( H , E ( H ) ) = ( 1 + O ( A − 1 + m − δ h ) ) p ∣ E ( H ) ∣ , \pi(H)=(1+O(A^{-1}))\pi(H,E(H))=(1+O(A^{-1}+m^{-\delta_h}))p^{|E(H)|}, π ( H ) = ( 1 + O ( A − 1 )) π ( H , E ( H )) = ( 1 + O ( A − 1 + m − δ h )) p ∣ E ( H ) ∣ , where δ h = 1 / ( h 2 h + 1 ) \delta_h=1/(h2^{h+1}) δ h = 1/ ( h 2 h + 1 )
Proof. First set C ∈ C ∈ C ( H ) C\in\mathcal{C}\in\mathcal{C}(H) C ∈ C ∈ C ( H ) ∣ C ∣ = k ≥ 3 |C|=k\ge3 ∣ C ∣ = k ≥ 3 E C = ∣ E ( H [ C ] ) ∣ E_C=|E(H[C])| E C = ∣ E ( H [ C ]) ∣ (10) , for m p 2 ≤ 1 mp^2\le1 m p 2 ≤ 1
π ( H , C ) π ( H , C [ { C } ↦ E ( H [ C ] ) ] ) ≍ m p k ( m p 2 ) E C = ( m − E C − k 2 E C − k p ) 2 E C − k = ( m f ( E C ) p ) 2 E C − k , (17) \frac{\pi(H,\mathcal{C})}{\pi(H,\mathcal{C}[\{C\}\mapsto E(H[C])])}
\asymp\frac{mp^k}{(mp^2)^{E_C}}
=\left(\frac{m^{-\frac{E_C-k}{2E_C-k}}}{p}\right)^{2E_C-k}
=\left(\frac{m^{f(E_C)}}{p}\right)^{2E_C-k}, \tag*{(17)} π ( H , C [{ C } ↦ E ( H [ C ])]) π ( H , C ) ≍ ( m p 2 ) E C m p k = p m − 2 E C − k E C − k 2 E C − k = ( p m f ( E C ) ) 2 E C − k , (17) where f ( x ) : = − x − 1 2 x − k = − 1 2 − k − 2 4 x − 2 k f(x):=-\frac{x-1}{2x-k}=-\frac{1}{2}-\frac{k-2}{4x-2k} f ( x ) := − 2 x − k x − 1 = − 2 1 − 4 x − 2 k k − 2 f f f
f ( x ) is increasing for x > k / 2 and f ( ( k 2 ) ) = − k + 1 2 k . f(x)\text{ is increasing for }x>k/2\quad\text{and}\quad f\left(\binom{k}{2}\right)=-\frac{k+1}{2k}. f ( x ) is increasing for x > k /2 and f ( ( 2 k ) ) = − 2 k k + 1 . Furthermore, note that by P2 the graph H [ C ] H[C] H [ C ] k ≥ 3 k\ge3 k ≥ 3
2 E C − k ≥ 2 ( k − 1 ) − k = k − 2 ≥ 1. (18) 2E_C-k\ge2(k-1)-k=k-2\ge1. \tag*{(18)} 2 E C − k ≥ 2 ( k − 1 ) − k = k − 2 ≥ 1. (18) The assumptions of ( i ) (i) ( i ) E C = ( k 2 ) E_C=\binom{k}{2} E C = ( 2 k ) p ≤ m − k + 1 2 k = m f ( ( k 2 ) ) p\le m^{-\frac{k+1}{2k}}=m^{f(\binom{k}{2})} p ≤ m − 2 k k + 1 = m f ( ( 2 k ) )
m f ( E C ) p ≥ 1. \frac{m^{f(E_C)}}{p}\ge1. p m f ( E C ) ≥ 1. Consequently, ( i ) (i) ( i ) (17) and (18) .
Now let us consider the case ( i i ) (ii) ( ii ) E C ≤ ( k 2 ) E_C\le\binom{k}{2} E C ≤ ( 2 k ) P2 H [ C ] H[C] H [ C ] E C ≥ k − 1 > k / 2 E_C\ge k-1>k/2 E C ≥ k − 1 > k /2 k ≥ 3 k\ge3 k ≥ 3 f f f p ≥ m − k + 1 2 k = m f ( ( k 2 ) ) p\ge m^{-\frac{k+1}{2k}}=m^{f(\binom{k}{2})} p ≥ m − 2 k k + 1 = m f ( ( 2 k ) )
m f ( E C ) p ≤ m f ( ( k 2 ) ) p ≤ 1. \frac{m^{f(E_C)}}{p}\le\frac{m^{f(\binom{k}{2})}}{p}\le1. p m f ( E C ) ≤ p m f ( ( 2 k ) ) ≤ 1. Applying this and (18) to (17) , we get ( i i ) (ii) ( ii )
Under the assumptions from ( i i i ) (iii) ( iii ) p ≥ A m f ( ( h 2 ) ) p\ge Am^{f(\binom{h}{2})} p ≥ A m f ( ( 2 h ) ) h = ∣ V ( H ) ∣ h=|V(H)| h = ∣ V ( H ) ∣ C ∈ C ( H ) ∖ { E ( H ) } C\in\mathcal{C}(H)\setminus\{E(H)\} C ∈ C ( H ) ∖ { E ( H )} C ∈ C C\in\mathcal{C} C ∈ C k = ∣ C ∣ k=|C| k = ∣ C ∣ 3 ≤ k ≤ h 3\le k\le h 3 ≤ k ≤ h f f f
m f ( E C ) p ≤ m f ( ( k 2 ) ) A m f ( ( h 2 ) ) ≤ A − 1 . \frac{m^{f(E_C)}}{p}\le\frac{m^{f(\binom{k}{2})}}{Am^{f(\binom{h}{2})}}\le A^{-1}. p m f ( E C ) ≤ A m f ( ( 2 h ) ) m f ( ( 2 k ) ) ≤ A − 1 . Therefore, applying (17) and (18) inductively for each C C C ∣ C ∣ ≥ 3 |C| \ge3 ∣ C ∣ ≥ 3 C ∈ C ( H ) ∖ { E ( H ) } C \in\mathcal{C}(H) \setminus\{E(H)\} C ∈ C ( H ) ∖ { E ( H )} A ≥ 1 A \ge1 A ≥ 1
π ( H , C ) ≲ A − 1 π ( H , E ( H ) ) . \pi(H,C) \lesssim A^{-1}\pi(H,E(H)). π ( H , C ) ≲ A − 1 π ( H , E ( H )) . This combined with (8) proves the first equality in ( i i i ) (iii) ( iii ) (11) .
The next lemma allows for a more refined comparison between the probability of a clique cover C C C C C C
Lemma 4.12. Let C ∈ C ( H ) C \in\mathcal{C}(H) C ∈ C ( H ) simple clique cover of H H H C 1 , C 2 ∈ C C_1,C_2 \in C C 1 , C 2 ∈ C ∣ C 1 ∩ C 2 ∣ = 1 |C_1 \cap C_2|=1 ∣ C 1 ∩ C 2 ∣ = 1
E ( C 1 , C 2 ) : = { e ∈ E ( H ) : ∣ e ∩ C 1 ∣ = ∣ e ∩ C 2 ∣ = 1 and C 1 ∩ C 2 ∩ e = ∅ } . \mathcal{E}(C_1,C_2):=\{e\in E(H): |e\cap C_1|=|e\cap C_2|=1\ \text{and}\ C_1\cap C_2\cap e=\varnothing\}. E ( C 1 , C 2 ) := { e ∈ E ( H ) : ∣ e ∩ C 1 ∣ = ∣ e ∩ C 2 ∣ = 1 and C 1 ∩ C 2 ∩ e = ∅ } . If p ≤ m − k + 1 2 k p\le m^{-\frac{k+1}{2k}} p ≤ m − 2 k k + 1 ∣ E ( C 1 , C 2 ) ∣ ≥ ( k − 1 ) / 2 |\mathcal{E}(C_1,C_2)|\ge(k-1)/2 ∣ E ( C 1 , C 2 ) ∣ ≥ ( k − 1 ) /2 k ≥ 1 k\ge1 k ≥ 1 C ′ ∈ C ( H ) C'\in\mathcal{C}(H) C ′ ∈ C ( H )
(i) π ( H , C ) ≲ π ( H , C ′ ) \pi(H,C)\lesssim\pi(H,C') π ( H , C ) ≲ π ( H , C ′ )
(ii) ∃ C ′ ∈ C ′ C 1 ∪ C 2 ⊆ C ′ \exists C'\in\mathcal{C}'\ C_1\cup C_2\subseteq C' ∃ C ′ ∈ C ′ C 1 ∪ C 2 ⊆ C ′
(iii) ∣ C ′ ∣ ≤ ∣ C ∣ − 1 |C'|\le|C|-1 ∣ C ′ ∣ ≤ ∣ C ∣ − 1
For an example of a configuration described in the lemma see Figure 4 .
Figure 4. An example of E ( C 1 , C 2 ) = { e 1 , e 2 , e 3 , e 4 } \mathcal{E}(C_1,C_2)=\{e_1,e_2,e_3,e_4\} E ( C 1 , C 2 ) = { e 1 , e 2 , e 3 , e 4 } C e i C_{e_i} C e i i = 1 , 2 , 3 , 4 i=1,2,3,4 i = 1 , 2 , 3 , 4
Proof. For each e ∈ E : = E ( C 1 , C 2 ) e\in\mathcal{E}:=\mathcal{E}(C_1,C_2) e ∈ E := E ( C 1 , C 2 ) C e C_e C e e e e C C C P1 the set C e C_e C e e ∈ E ( H ) e\in E(H) e ∈ E ( H ) e ↦ C e e\mapsto C_e e ↦ C e e ∈ E e\in\mathcal{E} e ∈ E e 1 , e 2 ∈ E e_1,e_2\in\mathcal{E} e 1 , e 2 ∈ E C e 1 = C e 2 C_{e_1}=C_{e_2} C e 1 = C e 2 C e 1 C_{e_1} C e 1 C 1 C_1 C 1 C 2 C_2 C 2 Figure 4 ). This contradicts P1 . Moreover, again by P1 , each C e C_e C e C i C_i C i i = 1 , 2 i=1,2 i = 1 , 2 C E = C 1 ∪ C 2 ∪ ⋃ e ∈ E C e \mathcal{C}_{\mathcal{E}}=\mathcal{C}_1\cup\mathcal{C}_2\cup\bigcup_{e\in\mathcal{E}}\mathcal{C}_e C E = C 1 ∪ C 2 ∪ ⋃ e ∈ E C e C E = { C 1 , C 2 } ∪ { C e , e ∈ E } \mathcal{C}_{\mathcal{E}}=\{\mathcal{C}_1,\mathcal{C}_2\}\cup\{\mathcal{C}_e,e\in\mathcal{E}\} C E = { C 1 , C 2 } ∪ { C e , e ∈ E }
∣ C E ∣ = ∣ C 1 ∪ C 2 ∪ ⋃ e ∈ E C e ∣ ≤ ∣ C 1 ∣ + ∣ C 2 ∣ − 1 + ∑ e ∈ E ( ∣ C e ∣ − 2 ) = ∣ C 1 ∣ + ∣ C 2 ∣ − 1 − 2 ∣ E ∣ + ∑ e ∈ E ∣ C e ∣ . \begin{aligned}
\left|\mathcal{C}_{\mathcal{E}}\right|=\left|\mathcal{C}_1\cup\mathcal{C}_2\cup\bigcup_{e\in\mathcal{E}}\mathcal{C}_e\right|\leq|\mathcal{C}_1|+|\mathcal{C}_2|-1+\sum_{e\in\mathcal{E}}(|\mathcal{C}_e|-2) \\
&=|\mathcal{C}_1|+|\mathcal{C}_2|-1-2|\mathcal{E}|+\sum_{e\in\mathcal{E}}|\mathcal{C}_e|.
\end{aligned} ∣ C E ∣ = C 1 ∪ C 2 ∪ e ∈ E ⋃ C e ≤ ∣ C 1 ∣ + ∣ C 2 ∣ − 1 + e ∈ E ∑ ( ∣ C e ∣ − 2 ) = ∣ C 1 ∣ + ∣ C 2 ∣ − 1 − 2∣ E ∣ + e ∈ E ∑ ∣ C e ∣. Let C ′ = C [ C E ↦ { C E } ] ∈ C ( H ) \mathcal{C}'=\mathcal{C}[\mathcal{C}_{\mathcal{E}}\mapsto\{\mathcal{C}_{\mathcal{E}}\}]\in\mathcal{C}(H) C ′ = C [ C E ↦ { C E }] ∈ C ( H ) p ≤ m − k + 1 2 k p\leq m^{-\frac{k+1}{2k}} p ≤ m − 2 k k + 1 m p 2 ≤ 1 mp^2\leq1 m p 2 ≤ 1 (10) gives us
π ( H , C ) π ( H , C ′ ) ≲ m p ∣ C 1 ∣ m p ∣ C 2 ∣ ∏ e ∈ E m p ∣ C e ∣ m p ∣ C E ∣ ≤ m ∣ E ∣ + 1 p 2 ∣ E ∣ + 1 ≤ ( m ∣ E ∣ + 1 2 ∣ E ∣ + 1 m − k + 1 2 k ) 2 ∣ E ∣ + 1 ≤ 1 , \frac{\pi(H,\mathcal{C})}{\pi(H,\mathcal{C}')} \lesssim\frac{mp^{|\mathcal{C}_1|}mp^{|\mathcal{C}_2|}\prod_{e\in\mathcal{E}}mp^{|\mathcal{C}_e|}}{mp^{|\mathcal{C}_{\mathcal{E}}|}}\leq m^{|\mathcal{E}|+1}p^{2|\mathcal{E}|+1}\leq\left(m^{\frac{|\mathcal{E}|+1}{2|\mathcal{E}|+1}}m^{-\frac{k+1}{2k}}\right)^{2|\mathcal{E}|+1}\leq1, π ( H , C ′ ) π ( H , C ) ≲ m p ∣ C E ∣ m p ∣ C 1 ∣ m p ∣ C 2 ∣ ∏ e ∈ E m p ∣ C e ∣ ≤ m ∣ E ∣ + 1 p 2∣ E ∣ + 1 ≤ ( m 2∣ E ∣ + 1 ∣ E ∣ + 1 m − 2 k k + 1 ) 2∣ E ∣ + 1 ≤ 1 , where the last inequality is a consequence of the assumption ∣ E ∣ ≥ ( k − 1 ) / 2 |\mathcal{E}|\geq(k-1)/2 ∣ E ∣ ≥ ( k − 1 ) /2 k + 1 2 k = k − 1 2 + 1 2 k − 1 2 + 1 \frac{k+1}{2k}=\frac{\frac{k-1}{2}+1}{2\frac{k-1}{2}+1} 2 k k + 1 = 2 2 k − 1 + 1 2 k − 1 + 1 f ( x ) = x + 1 2 x + 1 = 1 2 + 1 2 ( 2 x + 1 ) f(x)=\frac{x+1}{2x+1}=\frac{1}{2}+\frac{1}{2(2x+1)} f ( x ) = 2 x + 1 x + 1 = 2 1 + 2 ( 2 x + 1 ) 1 x > 0 x>0 x > 0 C \mathcal{C} C P2 , all graphs H [ C ] H[\mathcal{C}] H [ C ] C ∈ C E \mathcal{C}\in\mathcal{C}_{\mathcal{E}} C ∈ C E C E \mathcal{C}_{\mathcal{E}} C E C 1 \mathcal{C}_1 C 1 C 2 \mathcal{C}_2 C 2 H [ C E ] H[\mathcal{C}_{\mathcal{E}}] H [ C E ] C ′ \mathcal{C}' C ′ P2 . Obviously, C ′ \mathcal{C}' C ′ C ′ \mathcal{C}' C ′ P1 , then the lemma follows by Lemma 4.7 (ii) applied to C ′ \mathcal{C}' C ′
The following straightforward corollary of the previous lemma will play a key role when considering m p ≤ 1 mp\leq1 m p ≤ 1 [23 ] .
Corollary 4.1. Let m p ≤ 1 mp\leq1 m p ≤ 1 H H H
π ( H ) ≍ π ( H , { V ( H ) } ) ≍ m p ∣ V ( H ) ∣ . \pi(H)\asymp\pi(H,\{V(H)\})\asymp mp^{|V(H)|}. π ( H ) ≍ π ( H , { V ( H )}) ≍ m p ∣ V ( H ) ∣ . Proof. Let C ∈ C ( H ) \mathcal{C}\in\mathcal{C}(H) C ∈ C ( H ) ∣ C ∣ ≥ 2 |\mathcal{C}|\geq2 ∣ C ∣ ≥ 2 H H H C 1 \mathcal{C}_1 C 1 C 2 \mathcal{C}_2 C 2 C \mathcal{C} C k = 1 k=1 k = 1 C ′ ∈ C ( H ) \mathcal{C}'\in\mathcal{C}(H) C ′ ∈ C ( H ) ∣ C ′ ∣ ≤ ∣ C ∣ − 1 |\mathcal{C}'|\leq|\mathcal{C}|-1 ∣ C ′ ∣ ≤ ∣ C ∣ − 1 π ( H , C ) ≲ π ( H , C ′ ) \pi(H,\mathcal{C})\lesssim\pi(H,\mathcal{C}') π ( H , C ) ≲ π ( H , C ′ ) C ′ ′ ∈ C ( H ) \mathcal{C}''\in\mathcal{C}(H) C ′′ ∈ C ( H ) ∣ C ′ ′ ∣ = 1 |\mathcal{C}''|=1 ∣ C ′′ ∣ = 1 π ( H , C ) ≲ π ( H , C ′ ′ ) \pi(H,\mathcal{C})\lesssim\pi(H,\mathcal{C}'') π ( H , C ) ≲ π ( H , C ′′ ) H H H C ′ ′ = { V ( H ) } \mathcal{C}''=\{V(H)\} C ′′ = { V ( H )} C ∈ C ( H ) \mathcal{C}\in\mathcal{C}(H) C ∈ C ( H ) π ( H , C ) ≲ π ( H , { V ( H ) } ) \pi(H,\mathcal{C})\lesssim\pi(H,\{V(H)\}) π ( H , C ) ≲ π ( H , { V ( H )}) (9) and (10) .
Let us note that the assertion of Corollary 4.1 can be easily extended to disconnected graphs. Namely, the existence probability of such a graph is simply the product of the existence probabilities of its connected components.
In the next lemma, we will apply Lemma 4.12 to a special type of graphs, which constitutes a generalization of some graphs G t , I G_{t,I} G t , I (6) .
Lemma 4.13. Let H H H K 1 ⊕ j K 2 ⊆ H K_1\oplus_j K_2\subseteq H K 1 ⊕ j K 2 ⊆ H j ≥ 2 j\geq2 j ≥ 2 C ∈ C ( H ) \mathcal{C}\in\mathcal{C}(H) C ∈ C ( H ) C ∈ C \mathcal{C}\in\mathcal{C} C ∈ C p ≤ m − ∣ C ∣ + 1 2 ∣ C ∣ p\leq m^{-\frac{|\mathcal{C}|+1}{2|\mathcal{C}|}} p ≤ m − 2∣ C ∣ ∣ C ∣ + 1
∣ C ∩ V ( K 1 ⊕ j K 2 ) ∣ ≥ ∣ C ∣ + 1 2 , \left|\mathcal{C}\cap V(K_1\oplus_j K_2)\right|\geq\frac{|\mathcal{C}|+1}{2}, ∣ C ∩ V ( K 1 ⊕ j K 2 ) ∣ ≥ 2 ∣ C ∣ + 1 , then there exists a simple clique cover C ′ ∈ C ( H ) \mathcal{C}'\in\mathcal{C}(H) C ′ ∈ C ( H )
(i) ∃ C ′ ∈ C ′ \exists C'\in\mathcal{C}' ∃ C ′ ∈ C ′ C ∪ ( V ( K 1 ∩ K 2 ) ) ⊆ C ′ \mathcal{C}\cup(V(K_1\cap K_2))\subseteq C' C ∪ ( V ( K 1 ∩ K 2 )) ⊆ C ′
(ii) π ( H , C ) ≲ π ( H , C ′ ) \pi(H,\mathcal{C})\lesssim\pi(H,C') π ( H , C ) ≲ π ( H , C ′ )
Proof. If V ( K 1 ∩ K 2 ) ⊆ C V(K_1\cap K_2)\subseteq\mathcal{C} V ( K 1 ∩ K 2 ) ⊆ C v ∈ ( V ( K 1 ∩ K 2 ) ) ∖ C v\in(V(K_1\cap K_2))\setminus\mathcal{C} v ∈ ( V ( K 1 ∩ K 2 )) ∖ C w ∈ C ∩ V ( K 1 ⊕ j K 2 ) w\in\mathcal{C}\cap V(K_1\oplus_j K_2) w ∈ C ∩ V ( K 1 ⊕ j K 2 ) v v v V ( K 1 ⊕ j K 2 ) V(K_1\oplus_j K_2) V ( K 1 ⊕ j K 2 ) v w ∈ E ( H ) vw\in E(H) v w ∈ E ( H ) C \mathcal{C} C C 1 ∈ C C_1\in\mathcal{C} C 1 ∈ C v w vw v w ∣ C ∩ C 1 ∣ = 1 |\mathcal{C}\cap C_1|=1 ∣ C ∩ C 1 ∣ = 1 E ( C , C 1 ) \mathcal{E}(\mathcal{C},C_1) E ( C , C 1 ) Lemma 4.12 . Since v v v V ( K 1 ⊕ j K 2 ) V(K_1\oplus_j K_2) V ( K 1 ⊕ j K 2 ) Figure 5 )
Figure 5. The graph H H H K 1 ⊕ j K 2 K_1 \oplus_j K_2 K 1 ⊕ j K 2 C \mathcal{C} C C 1 C_1 C 1
E ( C , C 1 ) ⊇ { v z : z ≠ w , z ∈ C ∩ V ( K 1 ⊕ j K 2 ) } . \mathcal{E}(\mathcal{C},C_1)\supseteq\{vz:z\ne w,\ z\in\mathcal{C}\cap V(K_1\oplus_j K_2)\}. E ( C , C 1 ) ⊇ { v z : z = w , z ∈ C ∩ V ( K 1 ⊕ j K 2 )} . By the assumption ∣ C ∩ V ( K 1 ⊕ j K 2 ) ∣ ≥ ∣ C ∣ + 1 2 |\mathcal{C}\cap V(K_1\oplus_j K_2)|\ge\frac{|\mathcal{C}|+1}{2} ∣ C ∩ V ( K 1 ⊕ j K 2 ) ∣ ≥ 2 ∣ C ∣ + 1
∣ E ( C , C 1 ) ∣ ≥ ∣ C ∩ V ( K 1 ⊕ j K 2 ) ∣ − 1 ≥ ∣ C ∣ − 1 2 . |\mathcal{E}(\mathcal{C},C_1)|\ge|\mathcal{C}\cap V(K_1\oplus_j K_2)|-1\ge\frac{|\mathcal{C}|-1}{2}. ∣ E ( C , C 1 ) ∣ ≥ ∣ C ∩ V ( K 1 ⊕ j K 2 ) ∣ − 1 ≥ 2 ∣ C ∣ − 1 . Consequently, by Lemma 4.12 there is a simple C ′ ∈ C ( H ) C'\in\mathcal{C}(H) C ′ ∈ C ( H ) C ′ ∈ C ′ C'\in\mathcal{C}' C ′ ∈ C ′ C ∪ { v } ⊆ C ∪ C 1 ⊆ C ′ \mathcal{C}\cup\{v\}\subseteq\mathcal{C}\cup C_1\subseteq C' C ∪ { v } ⊆ C ∪ C 1 ⊆ C ′ π ( H , C ) ≲ π ( H , C ′ ) \pi(H,\mathcal{C})\lesssim\pi(H,C') π ( H , C ) ≲ π ( H , C ′ ) v ∈ ( V ( K 1 ∩ K 2 ) ) ∖ C v\in(V(K_1\cap K_2))\setminus\mathcal{C} v ∈ ( V ( K 1 ∩ K 2 )) ∖ C C ′ ′ ∈ C ( H ) C''\in\mathcal{C}(H) C ′′ ∈ C ( H ) C ′ ′ ∈ C ′ ′ C''\in\mathcal{C}'' C ′′ ∈ C ′′ C ∪ ( V ( K 1 ) ∩ V ( K 2 ) ) ⊆ C ′ ′ \mathcal{C}\cup(V(K_1)\cap V(K_2))\subseteq C'' C ∪ ( V ( K 1 ) ∩ V ( K 2 )) ⊆ C ′′ π ( H , C ) ≲ π ( H , C ′ ′ ) \pi(H,\mathcal{C})\lesssim\pi(H,C'') π ( H , C ) ≲ π ( H , C ′′ )
Remark 4.14. To understand the next two lemmas (Lemma 4.15 and Lemma 4.16 ), it is important to note that even if the value of π ( H 1 ) \pi(H_1) π ( H 1 ) π ( H 1 ) = ( 1 + o ( 1 ) ) π ( H 1 , E ( H 1 ) ) \pi(H_1)=(1+o(1))\pi(H_1,E(H_1)) π ( H 1 ) = ( 1 + o ( 1 )) π ( H 1 , E ( H 1 )) π ( H 1 ∪ H 2 ) \pi(H_1\cup H_2) π ( H 1 ∪ H 2 ) H 1 H_1 H 1 π ( H 1 ) = ( 1 + o ( 1 ) ) π ( H 1 , E ( H 1 ) ) \pi(H_1)=(1+o(1))\pi(H_1,E(H_1)) π ( H 1 ) = ( 1 + o ( 1 )) π ( H 1 , E ( H 1 )) π ( H 1 ∪ H 2 ) = ( 1 + o ( 1 ) ) π ( H 1 ∪ H 2 , { V ( H 1 ∪ H 2 ) } ) \pi(H_1\cup H_2)=(1+o(1))\pi(H_1\cup H_2,\{V(H_1\cup H_2)\}) π ( H 1 ∪ H 2 ) = ( 1 + o ( 1 )) π ( H 1 ∪ H 2 , { V ( H 1 ∪ H 2 )}) π ( H 1 ∪ H 2 ) \pi(H_1\cup H_2) π ( H 1 ∪ H 2 ) π ( H 1 ) \pi(H_1) π ( H 1 ) π ( H 2 ) \pi(H_2) π ( H 2 ) G ( n , p ) \mathcal{G}(n,\mathbf{p}) G ( n , p )
The next two lemmas allow us to estimate the probability of a union of graphs in terms of the probabilities of the graphs that constitute this sum. Together with Lemma 4.13, they will form the key part of the proof of Lemma 4.9 . Lemma 4.15 will also be useful for analysing the probabilities of the graphs G t G_t G t K a b K_{ab} K ab G t G_t G t
Lemma 4.15. Let m p ≥ 1 mp \ge1 m p ≥ 1 m p 2 ≤ 1 mp^2 \le1 m p 2 ≤ 1 H 1 , H 2 H_1,H_2 H 1 , H 2 K \mathbf{K} K r r r
(i) for a simple C ∈ C ( H 1 ⊕ 1 H 2 ) \mathcal{C} \in\mathcal{C}(H_1 \oplus_1 H_2) C ∈ C ( H 1 ⊕ 1 H 2 )
π ( H 1 ⊕ 1 H 2 , C ) ≲ π ( H 1 , C [ V ( H 1 ) ] ) π ( H 2 , C [ V ( H 2 ) ] ) . \pi(H_1 \oplus_1 H_2,\mathcal{C}) \lesssim\pi(H_1,\mathcal{C}[V(H_1)])\pi(H_2,\mathcal{C}[V(H_2)]). π ( H 1 ⊕ 1 H 2 , C ) ≲ π ( H 1 , C [ V ( H 1 )]) π ( H 2 , C [ V ( H 2 )]) . (ii) for a simple C ∈ C ( H 1 ⊕ 1 , 1 H 2 ) \mathcal{C} \in\mathcal{C}(H_1 \oplus_{1,1} H_2) C ∈ C ( H 1 ⊕ 1 , 1 H 2 )
π ( H 1 ⊕ 1 , 1 H 2 , C ) ≲ π ( H 1 , C [ V ( H 1 ) ] ) π ( H 2 , C [ V ( H 2 ) ] ) m p 2 . \pi(H_1 \oplus_{1,1} H_2,\mathcal{C}) \lesssim\frac{\pi(H_1,\mathcal{C}[V(H_1)])\pi(H_2,\mathcal{C}[V(H_2)])}{mp^2}. π ( H 1 ⊕ 1 , 1 H 2 , C ) ≲ m p 2 π ( H 1 , C [ V ( H 1 )]) π ( H 2 , C [ V ( H 2 )]) . (iii) if H = H 1 ⊕ i H 2 H = H_1 \oplus_i H_2 H = H 1 ⊕ i H 2 H = H 1 ⊕ i , j H 2 H = H_1 \oplus_{i,j} H_2 H = H 1 ⊕ i , j H 2 i , j ≥ 1 i,j \ge1 i , j ≥ 1 i + j ≤ r i+j \le r i + j ≤ r H 1 = K H_1 = \mathbf{K} H 1 = K C ∈ C ( H ) \mathcal{C} \in\mathcal{C}(H) C ∈ C ( H )
π ( H , C ) ≲ π ( H 1 , C [ V ( H 1 ) ] ) π ( H 2 , C [ V ( H 2 ) ] ) π ( H 1 [ V ( H 1 ∩ H 2 ) ] , C [ V ( H 1 ∩ H 2 ) ] ) . \pi(H,\mathcal{C}) \lesssim\frac{\pi(H_1,\mathcal{C}[V(H_1)])\pi(H_2,\mathcal{C}[V(H_2)])}{\pi(H_1[V(H_1 \cap H_2)],\mathcal{C}[V(H_1 \cap H_2)])}. π ( H , C ) ≲ π ( H 1 [ V ( H 1 ∩ H 2 )] , C [ V ( H 1 ∩ H 2 )]) π ( H 1 , C [ V ( H 1 )]) π ( H 2 , C [ V ( H 2 )]) . Proof. Let H = H 1 ⊕ i H 2 H = H_1 \oplus_i H_2 H = H 1 ⊕ i H 2 H = H 1 ⊕ i , j H 2 H = H_1 \oplus_{i,j} H_2 H = H 1 ⊕ i , j H 2 C ∈ C ( H ) \mathcal{C} \in\mathcal{C}(H) C ∈ C ( H ) V s = V ( H s ) V_s = V(H_s) V s = V ( H s ) s ∈ { 1 , 2 } s \in\{1,2\} s ∈ { 1 , 2 } C ∈ C C \in\mathcal{C} C ∈ C H 1 [ C ] H_1[C] H 1 [ C ] H 2 [ C ] H_2[C] H 2 [ C ] P2 implies ∣ C ∩ V ( H 1 ) ∩ V ( H 2 ) ∣ ≥ 1 |C \cap V(H_1) \cap V(H_2)| \ge1 ∣ C ∩ V ( H 1 ) ∩ V ( H 2 ) ∣ ≥ 1 H [ C ] H[C] H [ C ] C \mathcal{C} C C = C 1 ∪ ˙ C 2 ∪ ˙ C ∩ 1 ∪ ˙ C ∩ ≥ 2 \mathcal{C} = \mathcal{C}_1 \mathbin{\dot\cup} \mathcal{C}_2 \mathbin{\dot\cup} \mathcal{C}_{\cap1} \mathbin{\dot\cup} \mathcal{C}_{\cap\ge2} C = C 1 ∪ ˙ C 2 ∪ ˙ C ∩ 1 ∪ ˙ C ∩ ≥ 2
C 1 = { C ∈ C : C does not contain any edge of H 2 } , C 2 = { C ∈ C : C does not contain any edge of H 1 } , C ∩ 1 = { C ∈ C : C contains edges of both H 1 and H 2 , and ∣ C ∩ V 1 ∩ V 2 ∣ = 1 } , C ∩ ≥ 2 = { C ∈ C : C contains edges of both H 1 and H 2 , and ∣ C ∩ V 1 ∩ V 2 ∣ ≥ 2 } . \begin{aligned}
\mathcal{C}_1 &= \{C \in\mathcal{C} : C\text{ does not contain any edge of }H_2\},\\
\mathcal{C}_2 &= \{C \in\mathcal{C} : C\text{ does not contain any edge of }H_1\},\\
\mathcal{C}_{\cap1} &= \{C \in\mathcal{C} : C\text{ contains edges of both }H_1\text{ and }H_2,\text{ and }|C \cap V_1 \cap V_2| = 1\},\\
\mathcal{C}_{\cap\ge2} &= \{C \in\mathcal{C} : C\text{ contains edges of both }H_1\text{ and }H_2,\text{ and }|C \cap V_1 \cap V_2| \ge2\}.
\end{aligned} C 1 C 2 C ∩ 1 C ∩ ≥ 2 = { C ∈ C : C does not contain any edge of H 2 } , = { C ∈ C : C does not contain any edge of H 1 } , = { C ∈ C : C contains edges of both H 1 and H 2 , and ∣ C ∩ V 1 ∩ V 2 ∣ = 1 } , = { C ∈ C : C contains edges of both H 1 and H 2 , and ∣ C ∩ V 1 ∩ V 2 ∣ ≥ 2 } . (i) First, consider H = H 1 ⊕ 1 H 2 H = H_1 \oplus_1 H_2 H = H 1 ⊕ 1 H 2 C ∩ ≥ 2 = ∅ \mathcal{C}_{\cap\ge2} = \varnothing C ∩ ≥ 2 = ∅ ∣ V 1 ∩ V 2 ∣ = 1 |V_1 \cap V_2| = 1 ∣ V 1 ∩ V 2 ∣ = 1 (10) , for m p ≥ 1 mp \ge1 m p ≥ 1
π ( H , C ) ≲ ∏ C ∈ C 1 m p ∣ C ∣ ∏ C ∈ C 2 m p ∣ C ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − 1 ≤ ∏ i = 1 , 2 ( ∏ C ∈ C i m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V i ∣ ) ∏ C ∈ C ∩ 1 m p \begin{aligned}
\pi(H,\mathcal{C}) \lesssim\prod_{C \in\mathcal{C}_1}mp^{|C|}\prod_{C \in\mathcal{C}_2}mp^{|C|}\prod_{C \in\mathcal{C}_{\cap1}}mp^{|C \cap V_1|+|C \cap V_2|-1} \\
&\le\frac{\prod_{i=1,2}\left(\prod_{C \in\mathcal{C}_i}mp^{|C \cap V_i|}\prod_{C \in\mathcal{C}_{\cap1}}mp^{|C \cap V_i|}\right)}{\prod_{C \in\mathcal{C}_{\cap1}}mp}
\end{aligned} π ( H , C ) ≲ C ∈ C 1 ∏ m p ∣ C ∣ C ∈ C 2 ∏ m p ∣ C ∣ C ∈ C ∩ 1 ∏ m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − 1 ≤ ∏ C ∈ C ∩ 1 m p ∏ i = 1 , 2 ( ∏ C ∈ C i m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V i ∣ ) ≲ π ( H 1 , C [ V 1 ] ) π ( H 2 , C [ V 2 ] ) . \lesssim\pi(H_1,\mathcal{C}[V_1])\pi(H_2,\mathcal{C}[V_2]). ≲ π ( H 1 , C [ V 1 ]) π ( H 2 , C [ V 2 ]) . (ii) Now let H = H 1 ⊕ 1 , 1 H 2 H = H_1 \oplus_{1,1} H_2 H = H 1 ⊕ 1 , 1 H 2 ∣ V 1 ∩ V 2 ∣ = 2 |V_1 \cap V_2| = 2 ∣ V 1 ∩ V 2 ∣ = 2 C ∈ C C \in\mathcal{C} C ∈ C ∣ V 1 ∩ V 2 ∩ C ∣ ≥ 2 |V_1 \cap V_2 \cap C| \ge2 ∣ V 1 ∩ V 2 ∩ C ∣ ≥ 2 C \mathcal{C} C V 1 ∩ V 2 V_1 \cap V_2 V 1 ∩ V 2 P1 . Thus ∣ C ∩ ≥ 2 ∣ ≤ 1 |\mathcal{C}_{\cap\ge2}| \le1 ∣ C ∩ ≥ 2 ∣ ≤ 1 (10) , for m p ≥ 1 mp \ge1 m p ≥ 1 m p 2 ≤ 1 mp^{2} \le1 m p 2 ≤ 1
π ( H , C ) ≍ ∏ C ∈ C 1 m p ∣ C ∣ ∏ C ∈ C 2 m p ∣ C ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − 1 ∏ C ∈ C ∩ ≥ 2 m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − 2 \pi(H,\mathcal{C}) \asymp\prod_{C\in\mathcal{C}_{1}} mp^{|C|}\prod_{C\in\mathcal{C}_{2}} mp^{|C|}\prod_{C\in\mathcal{C}_{\cap1}} mp^{|C\cap V_{1}|+|C\cap V_{2}|-1}\prod_{C\in\mathcal{C}_{\cap\ge2}} mp^{|C\cap V_{1}|+|C\cap V_{2}|-2} π ( H , C ) ≍ C ∈ C 1 ∏ m p ∣ C ∣ C ∈ C 2 ∏ m p ∣ C ∣ C ∈ C ∩ 1 ∏ m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − 1 C ∈ C ∩ ≥ 2 ∏ m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − 2 ≲ ∏ i = 1 , 2 ( ∏ C ∈ C i m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ ≥ 2 m p ∣ C ∩ V i ∣ ) ∏ C ∈ C ∩ 1 m p ∏ C ∈ C ∩ ≥ 2 m p 2 \lesssim\frac{\prod_{i=1,2}\left(\prod_{C\in\mathcal{C}_{i}} mp^{|C\cap V_{i}|}\prod_{C\in\mathcal{C}_{\cap1}} mp^{|C\cap V_{i}|}\prod_{C\in\mathcal{C}_{\cap\ge2}} mp^{|C\cap V_{i}|}\right)}{\prod_{C\in\mathcal{C}_{\cap1}} mp\prod_{C\in\mathcal{C}_{\cap\ge2}} mp^{2}} ≲ ∏ C ∈ C ∩ 1 m p ∏ C ∈ C ∩ ≥ 2 m p 2 ∏ i = 1 , 2 ( ∏ C ∈ C i m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ ≥ 2 m p ∣ C ∩ V i ∣ ) ≲ π ( H 1 , C [ V 1 ] ) π ( H 2 , C [ V 2 ] ) m p 2 . \lesssim\frac{\pi(H_{1},\mathcal{C}[V_{1}])\pi(H_{2},\mathcal{C}[V_{2}])}{mp^{2}}. ≲ m p 2 π ( H 1 , C [ V 1 ]) π ( H 2 , C [ V 2 ]) . (iii) Now let us consider H = K ⊕ i H 2 H=\mathbf{K}\oplus_{i}H_{2} H = K ⊕ i H 2 H = K ⊕ i , j H 2 H=\mathbf{K}\oplus_{i,j}H_{2} H = K ⊕ i , j H 2 H 1 = K H_{1}=\mathbf{K} H 1 = K K [ V 1 ∩ V 2 ] \mathbf{K}[V_{1}\cap V_{2}] K [ V 1 ∩ V 2 ] i i i i + j i+j i + j P1 each edge of K [ V 1 ∩ V 2 ] \mathbf{K}[V_{1}\cap V_{2}] K [ V 1 ∩ V 2 ] C \mathcal{C} C { C ∩ V 1 ∩ V 2 : C ∈ C ∩ ≥ 2 } \{C\cap V_{1}\cap V_{2}:C\in\mathcal{C}_{\cap\ge2}\} { C ∩ V 1 ∩ V 2 : C ∈ C ∩ ≥ 2 }
We define a subset of C 1 \mathcal{C}_{1} C 1 C ∈ C C\in\mathcal{C} C ∈ C V 1 ∩ V 2 V_{1}\cap V_{2} V 1 ∩ V 2 H 2 H_{2} H 2
C 1 , ∩ ≥ 2 = { C ∈ C 1 : ∣ C ∩ V 1 ∩ V 2 ∣ ≥ 2 } . \mathcal{C}_{1,\cap\ge2}=\{C\in\mathcal{C}_{1}:|C\cap V_{1}\cap V_{2}|\ge2\}. C 1 , ∩ ≥ 2 = { C ∈ C 1 : ∣ C ∩ V 1 ∩ V 2 ∣ ≥ 2 } . Moreover, since H 1 = K H_{1}=\mathbf{K} H 1 = K C ∈ C 2 C\in\mathcal{C}_{2} C ∈ C 2 ∣ C ∩ V 1 ∣ ≤ 1 |C\cap V_{1}|\le1 ∣ C ∩ V 1 ∣ ≤ 1 C C C H 1 H_{1} H 1
C [ V 1 ∩ V 2 ] = { C ∩ V 1 ∩ V 2 : C ∈ C ∩ ≥ 2 ∪ C 1 , ∩ ≥ 2 } . \mathcal{C}[V_{1}\cap V_{2}]=\{C\cap V_{1}\cap V_{2}:C\in\mathcal{C}_{\cap\ge2}\cup\mathcal{C}_{1,\cap\ge2}\}. C [ V 1 ∩ V 2 ] = { C ∩ V 1 ∩ V 2 : C ∈ C ∩ ≥ 2 ∪ C 1 , ∩ ≥ 2 } . Combining this with (10) , we arrive at
π ( H , C ) ≲ ∏ C ∈ C 1 m p ∣ C ∩ V 1 ∣ ∏ C ∈ C 2 m p ∣ C ∩ V 2 ∣ \pi(H,\mathcal{C})\lesssim\prod_{C\in\mathcal{C}_{1}} mp^{|C\cap V_{1}|}\prod_{C\in\mathcal{C}_{2}} mp^{|C\cap V_{2}|} π ( H , C ) ≲ C ∈ C 1 ∏ m p ∣ C ∩ V 1 ∣ C ∈ C 2 ∏ m p ∣ C ∩ V 2 ∣ ∏ C ∈ C ∩ ≥ 2 m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − ∣ C ∩ V 1 ∩ V 2 ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − 1 = ∏ i = 1 , 2 ( ∏ C ∈ C i m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ ≥ 2 m p ∣ C ∩ V i ∣ ) ∏ C ∈ C ∩ ≥ 2 m p ∣ C ∩ V 1 ∩ V 2 ∣ ∏ C ∈ C ∩ 1 m p ∏ C ∈ C 1 , ∩ ≥ 2 m p ∣ C ∩ V 1 ∩ V 2 ∣ \begin{aligned}
\qquad\qquad\prod_{C\in\mathcal{C}_{\cap\ge2}} mp^{|C\cap V_{1}|+|C\cap V_{2}|-|C\cap V_{1}\cap V_{2}|}\prod_{C\in\mathcal{C}_{\cap1}} mp^{|C\cap V_{1}|+|C\cap V_{2}|-1} \\
&= \frac{\prod_{i=1,2}\left(\prod_{C\in\mathcal{C}_{i}} mp^{|C\cap V_{i}|}\prod_{C\in\mathcal{C}_{\cap1}} mp^{|C\cap V_{i}|}\prod_{C\in\mathcal{C}_{\cap\ge2}} mp^{|C\cap V_{i}|}\right)}{\prod_{C\in\mathcal{C}_{\cap\ge2}} mp^{|C\cap V_{1}\cap V_{2}|}\prod_{C\in\mathcal{C}_{\cap1}} mp}\prod_{C\in\mathcal{C}_{1,\cap\ge2}} mp^{|C\cap V_{1}\cap V_{2}|}
\end{aligned} C ∈ C ∩ ≥ 2 ∏ m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − ∣ C ∩ V 1 ∩ V 2 ∣ C ∈ C ∩ 1 ∏ m p ∣ C ∩ V 1 ∣ + ∣ C ∩ V 2 ∣ − 1 = ∏ C ∈ C ∩ ≥ 2 m p ∣ C ∩ V 1 ∩ V 2 ∣ ∏ C ∈ C ∩ 1 m p ∏ i = 1 , 2 ( ∏ C ∈ C i m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ 1 m p ∣ C ∩ V i ∣ ∏ C ∈ C ∩ ≥ 2 m p ∣ C ∩ V i ∣ ) C ∈ C 1 , ∩ ≥ 2 ∏ m p ∣ C ∩ V 1 ∩ V 2 ∣ ≲ π ( K , C [ K ] ) π ( H 2 , C [ V 2 ] ) π ( K [ V 1 ∩ V 2 ] , C [ V 1 ∩ V 2 ] ) ⋅ ∏ C ∈ C 1 , ∩ ≥ 2 m p ∣ C ∩ V 1 ∩ V 2 ∣ ∏ C ∈ C ∩ 1 m p . \lesssim\frac{\pi(\mathbf{K},\mathcal{C}[\mathbf{K}])\pi(H_{2},\mathcal{C}[V_{2}])}{\pi(\mathbf{K}[V_{1}\cap V_{2}],\mathcal{C}[V_{1}\cap V_{2}])}\cdot\frac{\prod_{C\in\mathcal{C}_{1,\cap\ge2}}mp^{|C\cap V_{1}\cap V_{2}|}}{\prod_{C\in\mathcal{C}_{\cap1}}mp}. ≲ π ( K [ V 1 ∩ V 2 ] , C [ V 1 ∩ V 2 ]) π ( K , C [ K ]) π ( H 2 , C [ V 2 ]) ⋅ ∏ C ∈ C ∩ 1 m p ∏ C ∈ C 1 , ∩ ≥ 2 m p ∣ C ∩ V 1 ∩ V 2 ∣ . We note that for C ∈ C 1 , ∩ ≥ 2 C\in\mathcal{C}_{1,\cap\ge2} C ∈ C 1 , ∩ ≥ 2 ∣ C ∩ V 1 ∩ V 2 ∣ ≥ 2 |C\cap V_{1}\cap V_{2}|\ge2 ∣ C ∩ V 1 ∩ V 2 ∣ ≥ 2 m p ≥ 1 mp\ge1 m p ≥ 1 m p 2 ≤ 1 mp^{2}\le1 m p 2 ≤ 1 □ \square □
Lemma 4.16. Assume m p 2 ≤ 1 mp^{2}\le1 m p 2 ≤ 1 H H H H ‾ \overline{H} H
H = K ⊕ j H ‾ or H = K ⊕ j 1 , j 2 H ‾ , for some j , j 1 , j 2 ≥ 2. H=\mathbf{K}\oplus_{j}\overline{H}\quad\text{or}\quad H=\mathbf{K}\oplus_{j_{1},j_{2}}\overline{H},\qquad\text{for some }j,j_{1},j_{2}\ge2. H = K ⊕ j H or H = K ⊕ j 1 , j 2 H , for some j , j 1 , j 2 ≥ 2. Moreover, fix a simple clique cover C ∈ C ( H ) \mathcal{C}\in\mathcal{C}(H) C ∈ C ( H )
∀ C ∈ C p ≥ m − ∣ C ∣ + 1 2 ∣ C ∣ \forall C\in\mathcal{C}\ p\ge m^{-\frac{|C|+1}{2|C|}} ∀ C ∈ C p ≥ m − 2∣ C ∣ ∣ C ∣ + 1
(ii) ∃ C ∈ C \exists C\in\mathcal{C} ∃ C ∈ C V ( K ∩ H ‾ ) ⊆ C V(K\cap\overline{H})\subseteq C V ( K ∩ H ) ⊆ C
then it holds that
n ∣ V ( H ) ∣ π ( H , C ) ≲ n r P ⋅ n ∣ V ( H ‾ ) ∣ π ( H ‾ ) Ψ . n^{|V(H)|}\pi(H,C)\lesssim\frac{n^r\mathbf{P}\cdot n^{|V(\overline{H})|}\pi(\overline{H})}{\Psi}. n ∣ V ( H ) ∣ π ( H , C ) ≲ Ψ n r P ⋅ n ∣ V ( H ) ∣ π ( H ) . Proof. Without loss of generality, we may set j 1 + j 2 = j j_1+j_2=j j 1 + j 2 = j H H H K [ V ( H ‾ ∩ K ) ] K[V(\overline{H}\cap K)] K [ V ( H ∩ K )] j j j
Let us first assume that (i) is fulfilled. Then by Lemma 4.11 (ii) we have
π ( H , C ) ≲ π ( H , E ( H ) ) . \pi(H,C)\lesssim\pi(H,E(H)). π ( H , C ) ≲ π ( H , E ( H )) . We consider two cases. For p ≤ m − r + 1 2 r p\leq m^{-\frac{r+1}{2r}} p ≤ m − 2 r r + 1 Lemma 4.11 (i) and Lemma 4.15 (iii) and then, in the last line, the fact that K [ V ( K ∩ H ‾ ) ] K[V(K\cap\overline{H})] K [ V ( K ∩ H )] j j j (14) , and (15) , we obtain
n ∣ V ( H ) ∣ π ( H , C ) ≲ n ∣ V ( H ) ∣ π ( H , E ( H ) ) ≲ n ∣ V ( H ) ∣ π ( H , { V ( K ) } ∪ ( E ( H ‾ ) ∖ E ( K ∩ H ‾ ) ) ) ≲ n r π ( K , { V ( K ) } ) n ∣ V ( H ‾ ) ∣ π ( H ‾ , { V ( K ∩ H ‾ ) } ∪ ( E ( H ‾ ) ∖ E ( K ∩ H ‾ ) ) ) n ∣ V ( K ∩ H ‾ ) ∣ π ( K [ V ( K ∩ H ‾ ) ] , { V ( K ∩ H ‾ ) } ) ≲ n r P ⋅ n ∣ V ( H ‾ ) ∣ π ( H ‾ ) Ψ j ≲ n r P ⋅ n ∣ V ( H ‾ ) ∣ π ( H ‾ ) Ψ . \begin{aligned}
n^{|V(H)|}\pi(H,C)&\lesssim n^{|V(H)|}\pi(H,E(H))\\
&\lesssim n^{|V(H)|}\pi\bigl(H,\{V(K)\}\cup(E(\overline{H})\setminus E(K\cap\overline{H}))\bigr)\\
&\lesssim\frac{n^r\pi(K,\{V(K)\})n^{|V(\overline{H})|}\pi\bigl(\overline{H},\{V(K\cap\overline{H})\}\cup(E(\overline{H})\setminus E(K\cap\overline{H}))\bigr)}{n^{|V(K\cap\overline{H})|}\pi\bigl(K[V(K\cap\overline{H})],\{V(K\cap\overline{H})\}\bigr)}\\
&\lesssim\frac{n^r\mathbf{P}\cdot n^{|V(\overline{H})|}\pi(\overline{H})}{\Psi_j}
\lesssim\frac{n^r\mathbf{P}\cdot n^{|V(\overline{H})|}\pi(\overline{H})}{\Psi}.
\end{aligned} n ∣ V ( H ) ∣ π ( H , C ) ≲ n ∣ V ( H ) ∣ π ( H , E ( H )) ≲ n ∣ V ( H ) ∣ π ( H , { V ( K )} ∪ ( E ( H ) ∖ E ( K ∩ H )) ) ≲ n ∣ V ( K ∩ H ) ∣ π ( K [ V ( K ∩ H )] , { V ( K ∩ H )} ) n r π ( K , { V ( K )}) n ∣ V ( H ) ∣ π ( H , { V ( K ∩ H )} ∪ ( E ( H ) ∖ E ( K ∩ H )) ) ≲ Ψ j n r P ⋅ n ∣ V ( H ) ∣ π ( H ) ≲ Ψ n r P ⋅ n ∣ V ( H ) ∣ π ( H ) . Similarly, for m − r + 1 2 r ≤ p ≤ m − 1 / 2 m^{-\frac{r+1}{2r}}\leq p\leq m^{-1/2} m − 2 r r + 1 ≤ p ≤ m − 1/2 Lemma 4.15 (iii), the fact that K [ V ( K ∩ H ‾ ) ] K[V(K\cap\overline{H})] K [ V ( K ∩ H )] j j j (14) , and (15) we obtain
n ∣ V ( H ) ∣ π ( H , C ) ≲ n ∣ V ( H ) ∣ π ( H , E ( H ) ) ≲ n r π ( K , E ( K ) ) n ∣ V ( H ‾ ) ∣ π ( H ‾ , E ( H ‾ ) ) n ∣ V ( K ∩ H ‾ ) ∣ π ( K [ V ( K ∩ H ‾ ) ] , E ( K [ V ( K ∩ H ‾ ) ] ) ) ≲ n r P ⋅ n ∣ V ( H ‾ ) ∣ π ( H ‾ ) Ψ j ≤ n r P ⋅ n ∣ V ( H ‾ ) ∣ π ( H ‾ ) Ψ . \begin{aligned}
n^{|V(H)|}\pi(H,C)&\lesssim n^{|V(H)|}\pi(H,E(H))\\
&\lesssim\frac{n^r\pi(K,E(K))n^{|V(\overline{H})|}\pi(\overline{H},E(\overline{H}))}{n^{|V(K\cap\overline{H})|}\pi\bigl(K[V(K\cap\overline{H})],E(K[V(K\cap\overline{H})])\bigr)}\\
&\lesssim\frac{n^r\mathbf{P}\cdot n^{|V(\overline{H})|}\pi(\overline{H})}{\Psi_j}
\leq\frac{n^r\mathbf{P}\cdot n^{|V(\overline{H})|}\pi(\overline{H})}{\Psi}.
\end{aligned} n ∣ V ( H ) ∣ π ( H , C ) ≲ n ∣ V ( H ) ∣ π ( H , E ( H )) ≲ n ∣ V ( K ∩ H ) ∣ π ( K [ V ( K ∩ H )] , E ( K [ V ( K ∩ H )]) ) n r π ( K , E ( K )) n ∣ V ( H ) ∣ π ( H , E ( H )) ≲ Ψ j n r P ⋅ n ∣ V ( H ) ∣ π ( H ) ≤ Ψ n r P ⋅ n ∣ V ( H ) ∣ π ( H ) . We shall now proceed to the case when assumption (ii) is fulfilled. In this case C [ V ( K ∩ H ‾ ) ] = { V ( K ∩ H ‾ ) } C[V(K\cap\overline{H})]=\{V(K\cap\overline{H})\} C [ V ( K ∩ H )] = { V ( K ∩ H )} p ≤ m − r + 1 2 r p\leq m^{-\frac{r+1}{2r}} p ≤ m − 2 r r + 1 n p ≥ 1 np\geq1 n p ≥ 1
For p ≤ m − r + 1 2 r p\leq m^{-\frac{r+1}{2r}} p ≤ m − 2 r r + 1 (14) and (15)
n ∣ V ( K ∩ H ‾ ) ∣ π ( K [ V ( K ∩ H ‾ ) ] , { V ( K ∩ H ‾ ) } ) ≍ Ψ j ≳ Ψ . n^{|V(K\cap\overline{H})|}\pi\bigl(K[V(K\cap\overline{H})],\{V(K\cap\overline{H})\}\bigr)\asymp\Psi_j\gtrsim\Psi. n ∣ V ( K ∩ H ) ∣ π ( K [ V ( K ∩ H )] , { V ( K ∩ H )} ) ≍ Ψ j ≳ Ψ. On the other hand, for n p ≥ 1 np\geq1 n p ≥ 1 (10) , (14) , and (15)
n ∣ V ( K ∩ H ‾ ) ∣ π ( K [ V ( K ∩ H ‾ ) ] , { V ( K ∩ H ‾ ) } ) ≍ n j m p j ≥ n 2 m p 2 ≍ Ψ 2 ≳ Ψ . n^{|V(K\cap\overline{H})|}\pi\bigl(K[V(K\cap\overline{H})],\{V(K\cap\overline{H})\}\bigr)\asymp n^jmp^j\geq n^2mp^2\asymp\Psi_2\gtrsim\Psi. n ∣ V ( K ∩ H ) ∣ π ( K [ V ( K ∩ H )] , { V ( K ∩ H )} ) ≍ n j m p j ≥ n 2 m p 2 ≍ Ψ 2 ≳ Ψ. Therefore, by Lemma 4.15 (iii)
n ∣ V ( H ) ∣ π ( H , C ) ≲ n r π ( K , C [ V ( K ) ] ) n ∣ V ( H ‾ ) ∣ π ( H ‾ , C [ V ( H ‾ ) ] ) n ∣ V ( K ∩ H ‾ ) ∣ π ( K [ V ( K ∩ H ‾ ) ] , { V ( K ∩ H ‾ ) } ) ≲ n r P ⋅ n ∣ V ( H ‾ ) ∣ π ( H ‾ ) Ψ . \begin{aligned}
n^{|V(H)|}\pi(H,C)&\lesssim\frac{n^r\pi(K,C[V(K)])n^{|V(\overline{H})|}\pi(\overline{H},C[V(\overline{H})])}{n^{|V(K\cap\overline{H})|}\pi\bigl(K[V(K\cap\overline{H})],\{V(K\cap\overline{H})\}\bigr)}\\
&\lesssim\frac{n^r\mathbf{P}\cdot n^{|V(\overline{H})|}\pi(\overline{H})}{\Psi}.
\end{aligned} n ∣ V ( H ) ∣ π ( H , C ) ≲ n ∣ V ( K ∩ H ) ∣ π ( K [ V ( K ∩ H )] , { V ( K ∩ H )} ) n r π ( K , C [ V ( K )]) n ∣ V ( H ) ∣ π ( H , C [ V ( H )]) ≲ Ψ n r P ⋅ n ∣ V ( H ) ∣ π ( H ) . Now we assume (ii), n p ≤ 1 np \le1 n p ≤ 1 m − r + 1 2 r ≤ p ≤ m − 1 / 2 m^{-\frac{r+1}{2r}} \le p \le m^{-1/2} m − 2 r r + 1 ≤ p ≤ m − 1/2 m p ≥ 1 mp \ge1 m p ≥ 1
C K = C ∩ V ( K ) and C H ˉ = C ∩ V ( H ˉ ) . C_K = C \cap V(\mathbf{K}) \quad\text{and} \quad C_{\bar H} = C \cap V(\bar{\mathbf{H}}). C K = C ∩ V ( K ) and C H ˉ = C ∩ V ( H ˉ ) . Recall that C C C C C C C \mathcal{C} C K [ V ( K ∩ H ˉ ) ] \mathbf{K}[V(\mathbf{K} \cap\bar{\mathbf{H}})] K [ V ( K ∩ H ˉ )] C \mathcal{C} C P1 . Let C ′ ∈ C ∖ { C } C' \in\mathcal{C} \setminus\{C\} C ′ ∈ C ∖ { C } C ′ C' C ′ K \mathbf{K} K H ˉ \bar{\mathbf{H}} H ˉ C ′ ∩ V ( K ) C' \cap V(\mathbf{K}) C ′ ∩ V ( K ) C [ K ] \mathcal{C}[\mathbf{K}] C [ K ] C ′ ∩ V ( H ˉ ) C' \cap V(\bar{\mathbf{H}}) C ′ ∩ V ( H ˉ ) C [ H ˉ ] \mathcal{C}[\bar{\mathbf{H}}] C [ H ˉ ] C ′ C' C ′ K \mathbf{K} K H ˉ \bar{\mathbf{H}} H ˉ P2 ∣ C ′ ∩ V ( K ∩ H ˉ ) ∣ = 1 |C' \cap V(\mathbf{K} \cap\bar{\mathbf{H}})| = 1 ∣ C ′ ∩ V ( K ∩ H ˉ ) ∣ = 1
∣ C ′ ∣ = ∣ C ′ ∩ V ( K ) ∣ + ∣ C ′ ∩ V ( H ˉ ) ∣ − ∣ C ′ ∩ V ( K ∩ H ˉ ) ∣ = ∣ C ′ ∩ V ( K ) ∣ + ∣ C ′ ∩ V ( H ˉ ) ∣ − 1. |C'| = |C' \cap V(\mathbf{K})| + |C' \cap V(\bar{\mathbf{H}})| - |C' \cap V(\mathbf{K} \cap\bar{\mathbf{H}})| = |C' \cap V(\mathbf{K})| + |C' \cap V(\bar{\mathbf{H}})| - 1. ∣ C ′ ∣ = ∣ C ′ ∩ V ( K ) ∣ + ∣ C ′ ∩ V ( H ˉ ) ∣ − ∣ C ′ ∩ V ( K ∩ H ˉ ) ∣ = ∣ C ′ ∩ V ( K ) ∣ + ∣ C ′ ∩ V ( H ˉ ) ∣ − 1. Let c c c C ∖ { C } \mathcal{C} \setminus\{C\} C ∖ { C } K \mathbf{K} K H ˉ \bar{\mathbf{H}} H ˉ (10) , since m p ≥ 1 mp \ge1 m p ≥ 1
π ( H , C ) ≲ m p ∣ C ∣ ∏ C ′ ∈ C ∖ { C } m p ∣ C ′ ∣ ≤ m p ∣ C ∣ ∏ C ′ ∈ C [ K ] ∖ { C K } m p ∣ C ′ ∣ ∏ C ′ ′ ∈ C [ H ˉ ] ∖ { C H ˉ } m p ∣ C ′ ′ ∣ ( m p ) c ≤ m p ∣ C ∣ ∏ C ′ ∈ C [ K ] ∖ { C K } m p ∣ C ′ ∣ ∏ C ′ ′ ∈ C [ H ˉ ] ∖ { C H ˉ } m p ∣ C ′ ′ ∣ . (19) \begin{aligned}
\pi(H,\mathcal{C}) \lesssim mp^{|C|} \prod_{C' \in\mathcal{C}\setminus\{C\}} mp^{|C'|} \\
&\le mp^{|C|}\frac{\prod_{C' \in\mathcal{C}[\mathbf{K}]\setminus\{C_K\}}mp^{|C'|}\prod_{C'' \in\mathcal{C}[\bar{\mathbf{H}}]\setminus\{C_{\bar H}\}}mp^{|C''|}}{(mp)^c} \\
&\le mp^{|C|}\prod_{C' \in\mathcal{C}[\mathbf{K}]\setminus\{C_K\}}mp^{|C'|}\prod_{C'' \in\mathcal{C}[\bar{\mathbf{H}}]\setminus\{C_{\bar H}\}}mp^{|C''|}. \tag*{(19)}
\end{aligned} π ( H , C ) ≲ m p ∣ C ∣ C ′ ∈ C ∖ { C } ∏ m p ∣ C ′ ∣ ≤ m p ∣ C ∣ ( m p ) c ∏ C ′ ∈ C [ K ] ∖ { C K } m p ∣ C ′ ∣ ∏ C ′′ ∈ C [ H ˉ ] ∖ { C H ˉ } m p ∣ C ′′ ∣ ≤ m p ∣ C ∣ C ′ ∈ C [ K ] ∖ { C K } ∏ m p ∣ C ′ ∣ C ′′ ∈ C [ H ˉ ] ∖ { C H ˉ } ∏ m p ∣ C ′′ ∣ . (19) Let us consider the first product in (19) . By (10) we get
∏ C ′ ∈ C [ K ] ∖ { C K } m p ∣ C ′ ∣ = ∏ C ′ ∈ C [ K ] ∖ { C K } m p ∣ C ′ ∣ ∏ { i , j } ∈ C K , i ≠ j m p 2 ( m p 2 ) ( ∣ C K ∣ 2 ) ≲ π ( K , ( C [ K ] ∖ { C K } ) ∪ E [ K [ C K ] ] ) p ( ∣ C K ∣ 2 ) ≤ π ( K ) p ( ∣ C K ∣ 2 ) . \prod_{C' \in\mathcal{C}[\mathbf{K}]\setminus\{C_K\}}mp^{|C'|}
=
\frac{\prod_{C' \in\mathcal{C}[\mathbf{K}]\setminus\{C_K\}}mp^{|C'|}\prod_{\{i,j\}\in C_K,\,i\ne j}mp^2}{(mp^2)^{\binom{|C_K|}{2}}}
\lesssim\frac{\pi\left(\mathbf{K},(\mathcal{C}[\mathbf{K}]\setminus\{C_K\})\cup E[\mathbf{K}[C_K]]\right)}{\mathbf{p}^{\binom{|C_K|}{2}}}
\le\frac{\pi(\mathbf{K})}{\mathbf{p}^{\binom{|C_K|}{2}}}. C ′ ∈ C [ K ] ∖ { C K } ∏ m p ∣ C ′ ∣ = ( m p 2 ) ( 2 ∣ C K ∣ ) ∏ C ′ ∈ C [ K ] ∖ { C K } m p ∣ C ′ ∣ ∏ { i , j } ∈ C K , i = j m p 2 ≲ p ( 2 ∣ C K ∣ ) π ( K , ( C [ K ] ∖ { C K }) ∪ E [ K [ C K ]] ) ≤ p ( 2 ∣ C K ∣ ) π ( K ) . Regarding the latter product in (19) , in an analogous manner we get
∏ C ′ ′ ∈ C [ H ˉ ] ∖ { C H ˉ } m p ∣ C ′ ′ ∣ = ∏ C ′ ′ ∈ C [ H ˉ ] m p ∣ C ′ ′ ∣ m p ∣ C H ˉ ∣ ≲ π ( H ˉ , C [ H ˉ ] ) m p ∣ C H ˉ ∣ ≤ π ( H ˉ ) m p ∣ C H ˉ ∣ . \prod_{C'' \in\mathcal{C}[\bar{\mathbf{H}}]\setminus\{C_{\bar H}\}}mp^{|C''|}
=
\frac{\prod_{C''\in\mathcal{C}[\bar{\mathbf{H}}]}mp^{|C''|}}{mp^{|C_{\bar H}|}}
\lesssim\frac{\pi(\bar{\mathbf{H}},\mathcal{C}[\bar{\mathbf{H}}])}{mp^{|C_{\bar H}|}}
\le\frac{\pi(\bar{\mathbf{H}})}{mp^{|C_{\bar H}|}}. C ′′ ∈ C [ H ˉ ] ∖ { C H ˉ } ∏ m p ∣ C ′′ ∣ = m p ∣ C H ˉ ∣ ∏ C ′′ ∈ C [ H ˉ ] m p ∣ C ′′ ∣ ≲ m p ∣ C H ˉ ∣ π ( H ˉ , C [ H ˉ ]) ≤ m p ∣ C H ˉ ∣ π ( H ˉ ) . Applying the last two estimates to (19) and using the equality
∣ V ( K ) ∩ V ( H ˉ ) ∣ = ∣ ( C ∩ V ( K ) ) ∩ ( C ∩ V ( H ˉ ) ) ∣ = ∣ C H ˉ ∣ + ∣ C K ∣ − ∣ C ∣ , |V(\mathbf{K}) \cap V(\bar{\mathbf{H}})| = |(C \cap V(\mathbf{K})) \cap(C \cap V(\bar{\mathbf{H}}))| = |C_{\bar H}| + |C_K| - |C|, ∣ V ( K ) ∩ V ( H ˉ ) ∣ = ∣ ( C ∩ V ( K )) ∩ ( C ∩ V ( H ˉ )) ∣ = ∣ C H ˉ ∣ + ∣ C K ∣ − ∣ C ∣ , we arrive at
n ∣ V ( H ) ∣ π ( H , C ) ≲ n ∣ V ( H ˉ ) ∣ + r − ∣ V ( K ) ∩ V ( H ˉ ) ∣ m p ∣ C ∣ π ( K ) p ( ∣ C K ∣ 2 ) π ( H ˉ ) m p ∣ C H ˉ ∣ n^{|V(H)|}\pi(H,\mathcal{C}) \lesssim n^{|V(\bar H)|+r-|V(\mathbf{K})\cap V(\bar{\mathbf{H}})|}mp^{|C|}\frac{\pi(\mathbf{K})}{\mathbf{p}^{\binom{|C_K|}{2}}}\frac{\pi(\bar{\mathbf{H}})}{mp^{|C_{\bar H}|}} n ∣ V ( H ) ∣ π ( H , C ) ≲ n ∣ V ( H ˉ ) ∣ + r − ∣ V ( K ) ∩ V ( H ˉ ) ∣ m p ∣ C ∣ p ( 2 ∣ C K ∣ ) π ( K ) m p ∣ C H ˉ ∣ π ( H ˉ ) ≲ n r π ( K ) ⋅ n ∣ V ( H ˉ ) ∣ π ( H ˉ ) n ∣ C K ∣ p ( ∣ C K ∣ 2 ) ( n p ) ∣ C ∣ − ∣ C H ˉ ∣ \lesssim\frac{n^r\pi(\mathbf{K})\cdot n^{|V(\bar H)|}\pi(\bar{\mathbf{H}})}{n^{|C_K|}\mathbf{p}^{\binom{|C_K|}{2}}}(np)^{|C|-|C_{\bar H}|} ≲ n ∣ C K ∣ p ( 2 ∣ C K ∣ ) n r π ( K ) ⋅ n ∣ V ( H ˉ ) ∣ π ( H ˉ ) ( n p ) ∣ C ∣ − ∣ C H ˉ ∣ ≲ n r P ⋅ n ∣ V ( H ˉ ) ∣ π ( H ˉ ) Ψ , \lesssim\frac{n^r\mathbf{P}\cdot n^{|V(\bar H)|}\pi(\bar{\mathbf{H}})}{\Psi}, ≲ Ψ n r P ⋅ n ∣ V ( H ˉ ) ∣ π ( H ˉ ) , where in the last bound we used the assumption n p ≤ 1 np \le1 n p ≤ 1 (14) for i = ∣ C K ∣ i=|C_K| i = ∣ C K ∣ (15) . ▫
The next lemma completes the preparation for the proof of Lemma 4.9 .
Lemma 4.17. Let j − i ≥ 2 j-i \ge2 j − i ≥ 2 i ≥ 0 i \ge0 i ≥ 0 t = 1 t=1 t = 1 i ≥ 2 i \ge2 i ≥ 2 t = 2 , 3 t=2,3 t = 2 , 3 I ⊆ Γ I \subseteq\Gamma I ⊆ Γ ∣ I ∣ ≥ 2 |I| \ge2 ∣ I ∣ ≥ 2 G t , I G_{t,I} G t , I C ∈ C ( G t , I ) \mathcal{C} \in\mathcal{C}(G_{t,I}) C ∈ C ( G t , I )
∃ C ∈ C p ≤ m − ∣ C ∣ + 1 2 ∣ C ∣ , (20) \exists_{C \in\mathcal{C}}\ p \le m^{-\frac{|C|+1}{2|C|}}, \tag*{(20)} ∃ C ∈ C p ≤ m − 2∣ C ∣ ∣ C ∣ + 1 , (20) then there exists another simple clique cover C ′ \mathcal{C}' C ′ C ′ ∈ C ′ C' \in\mathcal{C}' C ′ ∈ C ′ a b ∈ I ab \in I ab ∈ I
(i) π ( G t , I , C ) ≲ π ( G t , I , C ′ ) \pi(G_{t,I},\mathcal{C}) \lesssim\pi(G_{t,I},\mathcal{C}') π ( G t , I , C ) ≲ π ( G t , I , C ′ )
(ii) V ( G t , I ∖ { a b } ∩ K a b ) ⊆ C ′ V(G_{t,I\setminus\{ab\}} \cap K_{ab}) \subseteq C' V ( G t , I ∖ { ab } ∩ K ab ) ⊆ C ′
(iii) G t , I ∖ { a b } G_{t,I\setminus\{ab\}} G t , I ∖ { ab }
Proof. We consider the following decompositions G t , I = H 1 ∪ H 2 G_{t,I}=H_1 \cup H_2 G t , I = H 1 ∪ H 2 H ′ = H 1 H'=H_1 H ′ = H 1 H ′ = H 2 H'=H_2 H ′ = H 2
G 1 , Γ = ( K 11 ⊕ j K 12 ) ∪ ( K 21 ⊕ j K 22 ) , G t , Γ = ( K 11 ⊕ i K 12 ) ∪ ( K 21 ⊕ i K 22 ) , for t = 2 , 3 , G 2 , Γ = ( K 12 ⊕ j − i K 22 ) ∪ ( K 11 ⊕ j − i K 21 ) , G 1 , { 11 , 12 , 21 } = ( K 11 ⊕ j K 12 ) ∪ ( K 11 ⊕ j − i K 21 ) , G t , { 11 , 12 , 21 } = ( K 11 ⊕ i K 12 ) ∪ ( K 11 ⊕ j − i K 21 ) , for t = 2 , 3 , G t , I = ( K y ⊕ x K z ) ∪ ( K y ⊕ x K z ) , for I = { y , z } , x ∈ { i , j , j − i } . (21) \begin{aligned}
G_{1,\Gamma} &= (K_{11} \oplus_j K_{12}) \cup(K_{21} \oplus_j K_{22}),\\
G_{t,\Gamma} &= (K_{11} \oplus_i K_{12}) \cup(K_{21} \oplus_i K_{22}), && \text{for } t=2,3,\\
G_{2,\Gamma} &= (K_{12} \oplus_{j-i} K_{22}) \cup(K_{11} \oplus_{j-i} K_{21}),\\
G_{1,\{11,12,21\}} &= (K_{11} \oplus_j K_{12}) \cup(K_{11} \oplus_{j-i} K_{21}),\\
G_{t,\{11,12,21\}} &= (K_{11} \oplus_i K_{12}) \cup(K_{11} \oplus_{j-i} K_{21}), && \text{for } t=2,3,\\
G_{t,I} &= (K_y \oplus_x K_z) \cup(K_y \oplus_x K_z), && \text{for } I=\{y,z\},\ x \in\{i,j,j-i\}. \tag*{(21)}
\end{aligned} G 1 , Γ G t , Γ G 2 , Γ G 1 , { 11 , 12 , 21 } G t , { 11 , 12 , 21 } G t , I = ( K 11 ⊕ j K 12 ) ∪ ( K 21 ⊕ j K 22 ) , = ( K 11 ⊕ i K 12 ) ∪ ( K 21 ⊕ i K 22 ) , = ( K 12 ⊕ j − i K 22 ) ∪ ( K 11 ⊕ j − i K 21 ) , = ( K 11 ⊕ j K 12 ) ∪ ( K 11 ⊕ j − i K 21 ) , = ( K 11 ⊕ i K 12 ) ∪ ( K 11 ⊕ j − i K 21 ) , = ( K y ⊕ x K z ) ∪ ( K y ⊕ x K z ) , for t = 2 , 3 , for t = 2 , 3 , for I = { y , z } , x ∈ { i , j , j − i } . (21) We note that, for all connected G t , I G_{t,I} G t , I ∣ I ∣ = 3 |I|=3 ∣ I ∣ = 3 G t , I ≅ G t , { 11 , 12 , 21 } G_{t,I} \cong G_{t,\{11,12,21\}} G t , I ≅ G t , { 11 , 12 , 21 } ∣ I ∣ = 3 |I|=3 ∣ I ∣ = 3 G t , { 11 , 12 , 21 } G_{t,\{11,12,21\}} G t , { 11 , 12 , 21 } H 1 ∩ H 2 H_1 \cap H_2 H 1 ∩ H 2
For C C C (20) and for G t , I = H 1 ∪ H 2 G_{t,I}=H_1 \cup H_2 G t , I = H 1 ∪ H 2
∃ l ∈ { 1 , 2 } ∣ C ∩ V ( H l ) ∣ ≥ ∣ C ∣ + 1 2 . (22) \exists_{l \in\{1,2\}}\ |C \cap V(H_l)| \ge\frac{|C|+1}{2}. \tag*{(22)} ∃ l ∈ { 1 , 2 } ∣ C ∩ V ( H l ) ∣ ≥ 2 ∣ C ∣ + 1 . (22) If C C C H 1 H_1 H 1 H 2 H_2 H 2 C ∩ V ( H 1 ) , C ∩ V ( H 2 ) ≠ ∅ C \cap V(H_1), C \cap V(H_2) \ne\varnothing C ∩ V ( H 1 ) , C ∩ V ( H 2 ) = ∅ C C C C ∩ V ( H 1 ∩ H 2 ) ≠ ∅ C \cap V(H_1 \cap H_2) \ne\varnothing C ∩ V ( H 1 ∩ H 2 ) = ∅
∣ C ∣ = ∣ C ∩ V ( H 1 ) ∣ + ∣ C ∩ V ( H 2 ) ∣ − ∣ C ∩ V ( H 1 ∩ H 2 ) ∣ ≤ ∣ C ∩ V ( H 1 ) ∣ + ∣ C ∩ V ( H 2 ) ∣ − 1 , |C| = |C \cap V(H_1)| + |C \cap V(H_2)| - |C \cap V(H_1 \cap H_2)| \le|C \cap V(H_1)| + |C \cap V(H_2)| - 1, ∣ C ∣ = ∣ C ∩ V ( H 1 ) ∣ + ∣ C ∩ V ( H 2 ) ∣ − ∣ C ∩ V ( H 1 ∩ H 2 ) ∣ ≤ ∣ C ∩ V ( H 1 ) ∣ + ∣ C ∩ V ( H 2 ) ∣ − 1 , which implies (22) .
We start by considering the most challenging case I = Γ I=\Gamma I = Γ t = 2 t=2 t = 2 G 2 , Γ G_{2,\Gamma} G 2 , Γ (21) , ∣ C ∩ H 1 ∣ = ∣ C ∩ V ( K 11 ⊕ i K 12 ) ∣ ≥ ( ∣ C ∣ + 1 ) / 2 |C \cap H_1|=|C \cap V(K_{11} \oplus_i K_{12})| \ge(|C|+1)/2 ∣ C ∩ H 1 ∣ = ∣ C ∩ V ( K 11 ⊕ i K 12 ) ∣ ≥ ( ∣ C ∣ + 1 ) /2 H = G 2 , Γ H=G_{2,\Gamma} H = G 2 , Γ K 11 ⊕ i K 12 ⊆ G 2 , Γ K_{11} \oplus_i K_{12} \subseteq G_{2,\Gamma} K 11 ⊕ i K 12 ⊆ G 2 , Γ C ′ \mathcal{C}' C ′ C ′ ∈ C ′ C' \in\mathcal{C}' C ′ ∈ C ′ π ( G 2 , Γ , C ) ≲ π ( G 2 , Γ , C ′ ) \pi(G_{2,\Gamma},\mathcal{C}) \lesssim\pi(G_{2,\Gamma},\mathcal{C}') π ( G 2 , Γ , C ) ≲ π ( G 2 , Γ , C ′ ) C ⊆ C ′ C \subseteq C' C ⊆ C ′ ∣ C ∣ ≤ ∣ C ′ ∣ |C| \le|C'| ∣ C ∣ ≤ ∣ C ′ ∣ p ≤ m − ∣ C ∣ + 1 2 ∣ C ∣ ≤ m − ∣ C ′ ∣ + 1 2 ∣ C ′ ∣ p \le m^{-\frac{|C|+1}{2|C|}} \le m^{-\frac{|C'|+1}{2|C'|}} p ≤ m − 2∣ C ∣ ∣ C ∣ + 1 ≤ m − 2∣ C ′ ∣ ∣ C ′ ∣ + 1 V ( K 11 ∩ K 12 ) ⊆ C ′ V(K_{11} \cap K_{12}) \subseteq C' V ( K 11 ∩ K 12 ) ⊆ C ′ (21) and clique cover C ′ \mathcal{C}' C ′ ∣ C ′ ∩ V ( K 12 ⊕ j K 22 ) ∣ ≥ ( ∣ C ′ ∣ + 1 ) / 2 |\mathcal{C}'\cap V(K_{12}\oplus_j K_{22})|\ge(|\mathcal{C}'|+1)/2 ∣ C ′ ∩ V ( K 12 ⊕ j K 22 ) ∣ ≥ ( ∣ C ′ ∣ + 1 ) /2 C ′ ′ \mathcal{C}'' C ′′ C ′ ′ ∈ C ′ ′ C''\in\mathcal{C}'' C ′′ ∈ C ′′ π ( G 2 , Γ , C ′ ) ≲ π ( G 2 , Γ , C ′ ′ ) \pi(G_{2,\Gamma},\mathcal{C}')\lesssim\pi(G_{2,\Gamma},\mathcal{C}'') π ( G 2 , Γ , C ′ ) ≲ π ( G 2 , Γ , C ′′ ) V ( K 11 ∩ K 12 ) ⊆ C ′ ⊆ C ′ ′ V(K_{11}\cap K_{12})\subseteq C'\subseteq C'' V ( K 11 ∩ K 12 ) ⊆ C ′ ⊆ C ′′ V ( K 12 ∩ K 22 ) ⊆ C ′ ′ V(K_{12}\cap K_{22})\subseteq C'' V ( K 12 ∩ K 22 ) ⊆ C ′′ C ′ ′ \mathcal{C}'' C ′′ C ′ ′ C'' C ′′
V ( K 12 ∩ G 2 , Γ ∖ { 12 } ) ⊆ C ′ ′ , π ( G 2 , Γ , C ) ≲ π ( G 2 , Γ , C ′ ′ ) . V(K_{12}\cap G_{2,\Gamma\setminus\{12\}})\subseteq C'',\qquad\pi(G_{2,\Gamma},\mathcal{C})\lesssim\pi(G_{2,\Gamma},\mathcal{C}''). V ( K 12 ∩ G 2 , Γ ∖ { 12 } ) ⊆ C ′′ , π ( G 2 , Γ , C ) ≲ π ( G 2 , Γ , C ′′ ) . Since G 2 , Γ ∖ { 12 } G_{2,\Gamma\setminus\{12\}} G 2 , Γ ∖ { 12 } t = 2 t=2 t = 2 I = Γ I=\Gamma I = Γ (21) . Note that in these cases, C ′ \mathcal{C}' C ′ C ′ ′ \mathcal{C}'' C ′′ ∣ I ∣ = 2 |I|=2 ∣ I ∣ = 2 ∣ I ∣ = 3 |I|=3 ∣ I ∣ = 3 I = { 11 , 12 , 21 } I=\{11,12,21\} I = { 11 , 12 , 21 } a b = 12 ab=12 ab = 12 a b = 21 ab=21 ab = 21 I = Γ I=\Gamma I = Γ t = 1 , 3 t=1,3 t = 1 , 3 a b = 12 ab=12 ab = 12 a b = 21 ab=21 ab = 21
Now we are ready to prove Lemma 4.9 .
Proof of Lemma 4.9 . We will apply Lemma 4.16 . For that we note that, for any ∣ I ∣ ≥ 2 |I|\ge2 ∣ I ∣ ≥ 2 a b ∈ I ab\in I ab ∈ I G t , I = K ⊕ x H ˉ G_{t,I}=K\oplus_x\bar{H} G t , I = K ⊕ x H ˉ K = K a b K=K_{ab} K = K ab H ˉ = G t , I ∖ { a b } \bar{H}=G_{t,I\setminus\{ab\}} H ˉ = G t , I ∖ { ab } x x x { i , j , j − i , ( i , j − i ) } \{i,j,j-i,(i,j-i)\} { i , j , j − i , ( i , j − i )} a b ab ab G t , I ∖ { a b } G_{t,I\setminus\{ab\}} G t , I ∖ { ab } Lemma 4.17 we have that for a clique cover C ∈ C ( G t , I ) \mathcal{C}\in\mathcal{C}(G_{t,I}) C ∈ C ( G t , I ) Lemma 4.16 is true or there exists a b ∈ I ab\in I ab ∈ I C ′ \mathcal{C}' C ′ C ′ ∈ C ′ C'\in\mathcal{C}' C ′ ∈ C ′ G t , I ∖ { a b } G_{t,I\setminus\{ab\}} G t , I ∖ { ab } π ( G t , I , C ) ≲ π ( G t , I , C ′ ) \pi(G_{t,I},\mathcal{C})\lesssim\pi(G_{t,I},\mathcal{C}') π ( G t , I , C ) ≲ π ( G t , I , C ′ ) C ′ C' C ′ Lemma 4.16 is fulfilled. Thus, Lemma 4.9 follows by Lemma 4.16 .
Decompositions of clique covers
In this section we consider clique covers of the graphs that are unions of graphs that pairwise intersect on at most one vertex.
Definition 4.18. A family { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ H H H partition of H H H ∀ i , j ∈ Λ , i ≠ j ∣ V ( H i ) ∩ V ( H j ) ∣ ≤ 1 \forall_{i,j\in\Lambda,i\ne j}\lvert V(H_i)\cap V(H_j)\rvert\le1 ∀ i , j ∈ Λ , i = j ∣ V ( H i ) ∩ V ( H j )∣ ≤ 1
(ii) E ( H ) = ⋃ i ∈ Λ E ( H i ) E(H)=\bigcup_{i\in\Lambda}E(H_i) E ( H ) = ⋃ i ∈ Λ E ( H i )
For any J ⊆ Λ J\subseteq\Lambda J ⊆ Λ
H J = ( V ( H ) , ⋃ i ∈ J E ( H i ) ) = H ∖ ⋃ i ∈ Λ ∖ J E ( H i ) , (23) H_J=\left(V(H),\bigcup_{i\in J}E(H_i)\right)=H\setminus\bigcup_{i\in\Lambda\setminus J}E(H_i), \tag*{(23)} H J = ( V ( H ) , i ∈ J ⋃ E ( H i ) ) = H ∖ i ∈ Λ ∖ J ⋃ E ( H i ) , (23) which is a graph created from H H H H i H_i H i i ∈ J i\in J i ∈ J H ∅ = ( V ( H ) , ∅ ) H_{\varnothing}=(V(H),\varnothing) H ∅ = ( V ( H ) , ∅ )
Later on, we will consider some special partitions of graphs, as e.g.,
for any graph G G G E ( G ) = { e 1 , … , e s } E(G)=\{e_1,\ldots,e_s\} E ( G ) = { e 1 , … , e s } { H i } i ∈ { 1 , … , s } \{H_i\}_{i\in\{1,\ldots,s\}} { H i } i ∈ { 1 , … , s } H i = G [ e i ] H_i=G[e_i] H i = G [ e i ]
for G 1 , 0 , 1 G_{1,0,1} G 1 , 0 , 1 G t , 1 , 2 G_{t,1,2} G t , 1 , 2 t = 1 , 2 t=1,2 t = 1 , 2 { K a b } a b ∈ Γ \{K_{ab}\}_{ab\in\Gamma} { K ab } ab ∈ Γ
for G 3 , 1 , j G_{3,1,j} G 3 , 1 , j { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ Λ = { 0 , 1 , 2 } \Lambda=\{0,1,2\} Λ = { 0 , 1 , 2 } H 0 = G { 11 , 21 } = K 11 ⊕ j − 1 K 21 H_0=G_{\{11,21\}}=K_{11}\oplus_{j-1}K_{21} H 0 = G { 11 , 21 } = K 11 ⊕ j − 1 K 21 H 1 = K 12 H_1=K_{12} H 1 = K 12 H 2 = K 22 H_2=K_{22} H 2 = K 22
For examples of partitions we refer the reader to Figure 6 .
Figure 6. Let r = 4 r=4 r = 4 H = G 2 , 1 , 2 ⊕ 0 N 4 H=G_{2,1,2}\oplus_0 N_4 H = G 2 , 1 , 2 ⊕ 0 N 4 partition with Λ = Γ \Lambda=\Gamma Λ = Γ H a b = K a b H_{ab}=K_{ab} H ab = K ab a b ∈ Γ ab\in\Gamma ab ∈ Γ H { 11 , 21 , 22 } H_{\{11,21,22\}} H { 11 , 21 , 22 } ( K a b ) a b ∈ { 11 , 21 , 22 } (K_{ab})_{ab\in\{11,21,22\}} ( K ab ) ab ∈ { 11 , 21 , 22 } G 3 , 1 , 3 G_{3,1,3} G 3 , 1 , 3 H 0 = K 11 ⊕ 2 K 12 H_0=K_{11}\oplus_2 K_{12} H 0 = K 11 ⊕ 2 K 12 H 1 = K 21 H_1=K_{21} H 1 = K 21 H 2 = K 22 H_2=K_{22} H 2 = K 22
Definition 4.19. Let { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ H H H C ∈ C ( H ) \mathcal{C}\in\mathcal{C}(H) C ∈ C ( H ) C \mathcal{C} C decomposable (in relation to the partition { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ i ∈ Λ i\in\Lambda i ∈ Λ C i ⊆ C \mathcal{C}_i\subseteq\mathcal{C} C i ⊆ C C ‾ i ⊆ C \overline{\mathcal{C}}_i\subseteq\mathcal{C} C i ⊆ C C i ∈ C ( H i ) \mathcal{C}_i\in\mathcal{C}(H_i) C i ∈ C ( H i ) C ‾ i ∈ C ( H Λ ∖ { i } ) \overline{\mathcal{C}}_i\in\mathcal{C}(H_{\Lambda\setminus\{i\}}) C i ∈ C ( H Λ ∖ { i } ) C i ∩ C ‾ i = ∅ \mathcal{C}_i\cap\overline{\mathcal{C}}_i=\varnothing C i ∩ C i = ∅ C \mathcal{C} C indecomposable . Figure 7 and Figure 8 illustrate several examples of graphs and their decomposable and indecomposable clique covers.
Figure 7. Let r = 3 r = 3 r = 3 H = G 2 , 1 , 2 ⊕ 0 N 4 H = G_{2,1,2} \oplus_0 N_4 H = G 2 , 1 , 2 ⊕ 0 N 4 { K a b } a b ∈ Γ \{\mathrm{K}_{ab}\}_{ab\in\Gamma} { K ab } ab ∈ Γ H ∅ H_{\varnothing} H ∅ C a b C_{ab} C ab a b ∈ Γ ab\in\Gamma ab ∈ Γ K 22 \mathrm{K}_{22} K 22 V ( K 22 ) V(\mathrm{K}_{22}) V ( K 22 ) H { 11 , 21 } H_{\{11,21\}} H { 11 , 21 }
Figure 8. Examples of indecomposable clique covers of H H H { H i } i ∈ { 1 , 2 } \{H_i\}_{i\in\{1,2\}} { H i } i ∈ { 1 , 2 } H i H_i H i i = 1 , 2 i=1,2 i = 1 , 2 K 4 K_4 K 4 V ( H 1 ) V(H_1) V ( H 1 ) V ( H 2 ) V(H_2) V ( H 2 ) C ∩ C_\cap C ∩
For J ⊆ Λ J\subseteq\Lambda J ⊆ Λ C ∈ C ( H ) \mathcal{C}\in\mathcal{C}(H) C ∈ C ( H ) decomposable to H J H_J H J Figure 7 ), if there are pairwise disjoint subfamilies C J \mathcal{C}_J C J C i \mathcal{C}_i C i i ∈ Λ ∖ J i\in\Lambda\setminus J i ∈ Λ ∖ J C \mathcal{C} C C J ∪ ⋃ i ∈ Λ ∖ J C i ⊆ C \mathcal{C}_J\cup\bigcup_{i\in\Lambda\setminus J}\mathcal{C}_i\subseteq\mathcal{C} C J ∪ ⋃ i ∈ Λ ∖ J C i ⊆ C C i ∈ C ( H i ) \mathcal{C}_i\in\mathcal{C}(H_i) C i ∈ C ( H i ) i ∈ Λ ∖ J i\in\Lambda\setminus J i ∈ Λ ∖ J C J ∈ C ( H J ) \mathcal{C}_J\in\mathcal{C}(H_J) C J ∈ C ( H J ) { H i } i ∈ J \{H_i\}_{i\in J} { H i } i ∈ J H J H_J H J C J ( H ) \mathcal{C}_J(H) C J ( H ) H H H H J H_J H J
In other words, C \mathcal{C} C C i \mathcal{C}_i C i C ‾ i \overline{\mathcal{C}}_i C i C \mathcal{C} C C i \mathcal{C}_i C i V ( H i ) V(H_i) V ( H i ) H i H_i H i C ‾ i \overline{\mathcal{C}}_i C i H ∖ E ( H i ) H\setminus E(H_i) H ∖ E ( H i ) C \mathcal{C} C H J H_J H J H i H_i H i i ∈ Λ ∖ J i\in\Lambda\setminus J i ∈ Λ ∖ J C \mathcal{C} C H ∅ H_{\varnothing} H ∅ C i ∈ C ( H i ) \mathcal{C}_i\in\mathcal{C}(H_i) C i ∈ C ( H i ) i ∈ Λ i\in\Lambda i ∈ Λ ⋃ i ∈ Λ C i ⊆ C \bigcup_{i\in\Lambda}\mathcal{C}_i\subseteq\mathcal{C} ⋃ i ∈ Λ C i ⊆ C C \mathcal{C} C C i \mathcal{C}_i C i C J \mathcal{C}_J C J
Lemma 4.20. Let H ( l ) H^{(l)} H ( l ) l = 1 , 2 l=1,2 l = 1 , 2 { H i ( l ) } i ∈ Λ \{H_i^{(l)}\}_{i\in\Lambda} { H i ( l ) } i ∈ Λ l = 1 , 2 l=1,2 l = 1 , 2 J ⊆ Λ J\subseteq\Lambda J ⊆ Λ
H J ( 1 ) ≅ H J ( 2 ) and ∀ i ∈ Λ ∖ J H i ( 1 ) ≅ H i ( 2 ) , (24) H_J^{(1)}\cong H_J^{(2)}\quad\text{and}\quad\forall_{i\in\Lambda\setminus J}H_i^{(1)}\cong H_i^{(2)}, \tag*{(24)} H J ( 1 ) ≅ H J ( 2 ) and ∀ i ∈ Λ ∖ J H i ( 1 ) ≅ H i ( 2 ) , (24) then
π ( H ( 1 ) , C J ( H ( 1 ) ) ) = π ( H ( 2 ) , C J ( H ( 2 ) ) ) . \pi\left(H^{(1)},\mathcal{C}_J(H^{(1)})\right)=\pi\left(H^{(2)},\mathcal{C}_J(H^{(2)})\right). π ( H ( 1 ) , C J ( H ( 1 ) ) ) = π ( H ( 2 ) , C J ( H ( 2 ) ) ) . Proof. Let
φ J : V ( H J ( 1 ) ) → V ( H J ( 2 ) ) and φ i : V ( H i ( 1 ) ) → V ( H i ( 2 ) ) , for i ∈ Λ ∖ J , \varphi_J: V(H_J^{(1)}) \to V(H_J^{(2)}) \quad\text{and}\quad\varphi_i: V(H_i^{(1)}) \to V(H_i^{(2)}), \quad\text{for } i\in\Lambda\setminus J, φ J : V ( H J ( 1 ) ) → V ( H J ( 2 ) ) and φ i : V ( H i ( 1 ) ) → V ( H i ( 2 ) ) , for i ∈ Λ ∖ J , be bijections of vertex sets implied by (24) , i.e., edge-preserving bijections of the respective vertex sets of H i ( l ) H_i^{(l)} H i ( l ) i ∈ Λ ∖ J i\in\Lambda\setminus J i ∈ Λ ∖ J H J ( l ) H_J^{(l)} H J ( l ) l = 1 , 2 l=1,2 l = 1 , 2
ϕ x : C ( H x ( 1 ) ) → C ( H x ( 2 ) ) , x ∈ { J } ∪ ( Λ ∖ J ) , \phi_x:\mathcal{C}(H_x^{(1)}) \to\mathcal{C}(H_x^{(2)}), \quad x\in\{J\}\cup(\Lambda\setminus J), ϕ x : C ( H x ( 1 ) ) → C ( H x ( 2 ) ) , x ∈ { J } ∪ ( Λ ∖ J ) , given by the formula
ϕ x ( { C 1 , … , C t } ) = { ϕ x ( C 1 ) , … , ϕ x ( C t ) } , x ∈ { J } ∪ ( Λ ∖ J ) , \phi_x(\{C_1,\ldots,C_t\})=\{\phi_x(C_1),\ldots,\phi_x(C_t)\}, \quad x\in\{J\}\cup(\Lambda\setminus J), ϕ x ({ C 1 , … , C t }) = { ϕ x ( C 1 ) , … , ϕ x ( C t )} , x ∈ { J } ∪ ( Λ ∖ J ) , for any C = { C 1 , … , C t } ∈ C ( H x ( 1 ) ) \mathcal{C}=\{C_1,\ldots,C_t\}\in\mathcal{C}(H_x^{(1)}) C = { C 1 , … , C t } ∈ C ( H x ( 1 ) ) (24) gives us P ( H x ( 1 ) ) ≃ P ( H x ( 2 ) ) \mathcal{P}(H_x^{(1)})\simeq\mathcal{P}(H_x^{(2)}) P ( H x ( 1 ) ) ≃ P ( H x ( 2 ) ) ≃ \simeq ≃ (13) . Therefore
∀ x ∈ { J } ∪ ( Λ ∖ J ) ∀ C ∈ C ( H x ( 1 ) ) C ≃ ϕ x ( C ) and P ( H x ( 1 ) ) ∖ C ≃ P ( H x ( 2 ) ) ∖ ϕ x ( C ) . (25) \forall x\in\{J\}\cup(\Lambda\setminus J)\ \forall\mathcal{C}\in\mathcal{C}(H_x^{(1)})\quad\mathcal{C}\simeq\phi_x(\mathcal{C})\quad\text{and}\quad\mathcal{P}(H_x^{(1)})\setminus\mathcal{C}\simeq\mathcal{P}(H_x^{(2)})\setminus\phi_x(\mathcal{C}). \tag*{(25)} ∀ x ∈ { J } ∪ ( Λ ∖ J ) ∀ C ∈ C ( H x ( 1 ) ) C ≃ ϕ x ( C ) and P ( H x ( 1 ) ) ∖ C ≃ P ( H x ( 2 ) ) ∖ ϕ x ( C ) . (25) Let C J ( H J ( l ) ) \mathcal{C}_J(H_J^{(l)}) C J ( H J ( l ) ) H J ( l ) H_J^{(l)} H J ( l ) ( H i ( l ) ) i ∈ J (H_i^{(l)})_{i\in J} ( H i ( l ) ) i ∈ J H ( l ) H^{(l)} H ( l ) H J ( l ) H_J^{(l)} H J ( l ) H J ( l ) H_J^{(l)} H J ( l ) H i ( l ) H_i^{(l)} H i ( l ) i ∈ Λ ∖ J i\in\Lambda\setminus J i ∈ Λ ∖ J
C J ( H ( l ) ) = ⋃ C J ∈ C J ( H J ( l ) ) ⋃ C i ∈ C ( H i ( l ) ) , i ∈ Λ ∖ J C ( H ( l ) , C J ∪ ⋃ i ∈ Λ ∖ J C i , ( P ( H J ( l ) ) ∖ C J ) ∪ ⋃ i ∈ Λ ∖ J ( P ( H i ( l ) ) ∖ C i ) ) . \mathcal{C}_J(H^{(l)})=
\bigcup_{\mathcal{C}_J\in\mathcal{C}_J(H_J^{(l)})}
\ \bigcup_{\mathcal{C}_i\in\mathcal{C}(H_i^{(l)}),\,i\in\Lambda\setminus J}
\mathcal{C}\left(H^{(l)},\mathcal{C}_J\cup\bigcup_{i\in\Lambda\setminus J}\mathcal{C}_i,\left(\mathcal{P}(H_J^{(l)})\setminus\mathcal{C}_J\right)\cup\bigcup_{i\in\Lambda\setminus J}\left(\mathcal{P}(H_i^{(l)})\setminus\mathcal{C}_i\right)\right). C J ( H ( l ) ) = C J ∈ C J ( H J ( l ) ) ⋃ C i ∈ C ( H i ( l ) ) , i ∈ Λ ∖ J ⋃ C H ( l ) , C J ∪ i ∈ Λ ∖ J ⋃ C i , ( P ( H J ( l ) ) ∖ C J ) ∪ i ∈ Λ ∖ J ⋃ ( P ( H i ( l ) ) ∖ C i ) . We will show that for a fixed l ∈ { 1 , 2 } l \in\{1,2\} l ∈ { 1 , 2 } l = 1 l=1 l = 1 C 1 \mathcal{C}_1 C 1 C 2 \mathcal{C}_2 C 2
C s , J ∈ C J ( H J ( 1 ) ) , C s , i ∈ C i ( H i ( 1 ) ) , i ∈ Λ ∖ J , s = 1 , 2 , \mathcal{C}_{s,J} \in\mathcal{C}_J(H_J^{(1)}), \qquad\mathcal{C}_{s,i} \in\mathcal{C}_i(H_i^{(1)}), \qquad i \in\Lambda\setminus J,\ s=1,2, C s , J ∈ C J ( H J ( 1 ) ) , C s , i ∈ C i ( H i ( 1 ) ) , i ∈ Λ ∖ J , s = 1 , 2 , and satisfying
∃ x ∈ { J } ∪ ( Λ ∖ J ) C 1 , x ≠ C 2 , x . \exists x\in\{J\}\cup(\Lambda\setminus J)\ \mathcal{C}_{1,x}\ne\mathcal{C}_{2,x}. ∃ x ∈ { J } ∪ ( Λ ∖ J ) C 1 , x = C 2 , x . Thus, without loss of generality, we may assume that there exists C ∈ C 1 , x ∖ C 2 , x C\in\mathcal{C}_{1,x}\setminus\mathcal{C}_{2,x} C ∈ C 1 , x ∖ C 2 , x x ∈ { J } ∪ ( Λ ∖ J ) x\in\{J\}\cup(\Lambda\setminus J) x ∈ { J } ∪ ( Λ ∖ J ) C ∈ C 1 , x C\in\mathcal{C}_{1,x} C ∈ C 1 , x C ∈ P ( H x ( 1 ) ) ∖ C 2 , x C\in\mathcal{P}(H_x^{(1)})\setminus\mathcal{C}_{2,x} C ∈ P ( H x ( 1 ) ) ∖ C 2 , x
C ∈ C 1 , J ∪ ⋃ i ∈ Λ ∖ J C 1 , i and C ∈ ( P ( H J ( 1 ) ) ∖ C 2 , J ) ∪ ⋃ i ∈ Λ ∖ J ( P ( H i ( 1 ) ) ∖ C 2 , i ) . C\in\mathcal{C}_{1,J}\cup\bigcup_{i\in\Lambda\setminus J}\mathcal{C}_{1,i}
\quad\text{and}\quad
C\in\left(\mathcal{P}(H_J^{(1)})\setminus\mathcal{C}_{2,J}\right)\cup\bigcup_{i\in\Lambda\setminus J}\left(\mathcal{P}(H_i^{(1)})\setminus\mathcal{C}_{2,i}\right). C ∈ C 1 , J ∪ i ∈ Λ ∖ J ⋃ C 1 , i and C ∈ ( P ( H J ( 1 ) ) ∖ C 2 , J ) ∪ i ∈ Λ ∖ J ⋃ ( P ( H i ( 1 ) ) ∖ C 2 , i ) . This implies that C C C C 1 \mathcal{C}_1 C 1 C 2 \mathcal{C}_2 C 2 C 1 \mathcal{C}_1 C 1 C 2 \mathcal{C}_2 C 2
Consequently, combining (25) , (8) and Lemma 4.4 we get
π ( H ( 1 ) , C J ( H ( 1 ) ) ) = ∑ C J ∈ C J ( H J ( 1 ) ) ∑ C i ∈ C ( H i ( 1 ) ) , i ∈ Λ ∖ J π ( H ( 1 ) , C J ∪ ⋃ i ∈ Λ ∖ J C i , ( P ( H J ( 1 ) ) ∖ C J ) ∪ ⋃ i ∈ Λ ∖ J ( P ( H i ( 1 ) ) ∖ C i ) ) = ∑ C J ∈ C J ( H J ( 1 ) ) ∑ C i ∈ C ( H i ( 1 ) ) , i ∈ Λ ∖ J π ( H ( 2 ) , ϕ J ( C J ) ∪ ⋃ i ∈ Λ ∖ J ϕ ( C i ) , ( P ( H J ( 2 ) ) ∖ ϕ ( C J ) ) ∪ ⋃ i ∈ Λ ∖ J ( P ( H i ( 2 ) ) ∖ ϕ ( C i ) ) ) = ∑ C J ∈ C J ( H J ( 2 ) ) ∑ C i ∈ C ( H i ( 2 ) ) , i ∈ Λ ∖ J π ( H ( 2 ) , C J ∪ ⋃ i ∈ Λ ∖ J C i , ( P ( H J ( 2 ) ) ∖ C J ) ∪ ⋃ i ∈ Λ ∖ J ( P ( H i ( 2 ) ) ∖ C i ) ) = π ( H ( 2 ) , C J ( H ( 2 ) ) ) , \begin{aligned}
\pi\left(H^{(1)},\mathcal{C}_J(H^{(1)})\right)
&=\sum_{\mathcal{C}_J\in\mathcal{C}_J(H_J^{(1)})}\sum_{\mathcal{C}_i\in\mathcal{C}(H_i^{(1)}),\,i\in\Lambda\setminus J}
\pi\left(H^{(1)},\mathcal{C}_J\cup\bigcup_{i\in\Lambda\setminus J}\mathcal{C}_i,\right.\\
&\qquad\left.\left(\mathcal{P}(H_J^{(1)})\setminus\mathcal{C}_J\right)\cup\bigcup_{i\in\Lambda\setminus J}\left(\mathcal{P}(H_i^{(1)})\setminus\mathcal{C}_i\right)\right)\\
&=\sum_{\mathcal{C}_J\in\mathcal{C}_J(H_J^{(1)})}\sum_{\mathcal{C}_i\in\mathcal{C}(H_i^{(1)}),\,i\in\Lambda\setminus J}
\pi\left(H^{(2)},\phi_J(\mathcal{C}_J)\cup\bigcup_{i\in\Lambda\setminus J}\phi(\mathcal{C}_i),\right.\\
&\qquad\left.\left(\mathcal{P}(H_J^{(2)})\setminus\phi(\mathcal{C}_J)\right)\cup\bigcup_{i\in\Lambda\setminus J}\left(\mathcal{P}(H_i^{(2)})\setminus\phi(\mathcal{C}_i)\right)\right)\\
&=\sum_{\mathcal{C}_J\in\mathcal{C}_J(H_J^{(2)})}\sum_{\mathcal{C}_i\in\mathcal{C}(H_i^{(2)}),\,i\in\Lambda\setminus J}
\pi\left(H^{(2)},\mathcal{C}_J\cup\bigcup_{i\in\Lambda\setminus J}\mathcal{C}_i,\right.\\
&\qquad\left.\left(\mathcal{P}(H_J^{(2)})\setminus\mathcal{C}_J\right)\cup\bigcup_{i\in\Lambda\setminus J}\left(\mathcal{P}(H_i^{(2)})\setminus\mathcal{C}_i\right)\right)\\
&=\pi\left(H^{(2)},\mathcal{C}_J(H^{(2)})\right),
\end{aligned} π ( H ( 1 ) , C J ( H ( 1 ) ) ) = C J ∈ C J ( H J ( 1 ) ) ∑ C i ∈ C ( H i ( 1 ) ) , i ∈ Λ ∖ J ∑ π H ( 1 ) , C J ∪ i ∈ Λ ∖ J ⋃ C i , ( P ( H J ( 1 ) ) ∖ C J ) ∪ i ∈ Λ ∖ J ⋃ ( P ( H i ( 1 ) ) ∖ C i ) = C J ∈ C J ( H J ( 1 ) ) ∑ C i ∈ C ( H i ( 1 ) ) , i ∈ Λ ∖ J ∑ π H ( 2 ) , ϕ J ( C J ) ∪ i ∈ Λ ∖ J ⋃ ϕ ( C i ) , ( P ( H J ( 2 ) ) ∖ ϕ ( C J ) ) ∪ i ∈ Λ ∖ J ⋃ ( P ( H i ( 2 ) ) ∖ ϕ ( C i ) ) = C J ∈ C J ( H J ( 2 ) ) ∑ C i ∈ C ( H i ( 2 ) ) , i ∈ Λ ∖ J ∑ π H ( 2 ) , C J ∪ i ∈ Λ ∖ J ⋃ C i , ( P ( H J ( 2 ) ) ∖ C J ) ∪ i ∈ Λ ∖ J ⋃ ( P ( H i ( 2 ) ) ∖ C i ) = π ( H ( 2 ) , C J ( H ( 2 ) ) ) , as required.
The next result will be essential for obtaining sharp estimates of subgraph probabilities in the case m p 2 ≥ 1 mp^2\ge1 m p 2 ≥ 1
Lemma 4.21. Let m p 2 ≥ 1 mp^2\ge1 m p 2 ≥ 1 m p 3 ≤ 1 mp^3\le1 m p 3 ≤ 1 H 1 H_1 H 1 H 2 H_2 H 2
π ( H 1 ∪ H 2 ) = π ( H 1 ) π ( H 2 ) π ( H 1 ∩ H 2 ) + O ( m p 3 e − 2 m p 2 ) . \pi(H_1\cup H_2)=\frac{\pi(H_1)\pi(H_2)}{\pi(H_1\cap H_2)}+O(mp^3e^{-2mp^2}). π ( H 1 ∪ H 2 ) = π ( H 1 ∩ H 2 ) π ( H 1 ) π ( H 2 ) + O ( m p 3 e − 2 m p 2 ) . Proof. Let G ( 1 ) = ( H 1 ∪ H 2 ) ⊕ 0 ( H 1 ∩ H 2 ) G^{(1)}=(H_1\cup H_2)\oplus_0(H_1\cap H_2) G ( 1 ) = ( H 1 ∪ H 2 ) ⊕ 0 ( H 1 ∩ H 2 ) G ( 2 ) = H 1 ⊕ 0 H 2 G^{(2)}=H_1\oplus_0 H_2 G ( 2 ) = H 1 ⊕ 0 H 2 G ( 1 ) G^{(1)} G ( 1 ) G ( 2 ) G^{(2)} G ( 2 ) E ( G ( 1 ) ) = { e 1 ( 1 ) , … , e l ( 1 ) } E(G^{(1)})=\{e^{(1)}_1,\ldots,e^{(1)}_l\} E ( G ( 1 ) ) = { e 1 ( 1 ) , … , e l ( 1 ) } E ( G ( 2 ) ) = { e 1 ( 2 ) , … , e l ( 2 ) } E(G^{(2)})=\{e^{(2)}_1,\ldots,e^{(2)}_l\} E ( G ( 2 ) ) = { e 1 ( 2 ) , … , e l ( 2 ) } j = 1 , 2 j=1,2 j = 1 , 2 { H ˉ i ( j ) } i ∈ { 1 , … , l } \{\bar H^{(j)}_i\}_{i\in\{1,\ldots,l\}} { H ˉ i ( j ) } i ∈ { 1 , … , l } H ˉ i ( j ) \bar H^{(j)}_i H ˉ i ( j ) G ( j ) G^{(j)} G ( j ) e i ( j ) e^{(j)}_i e i ( j ) H ˉ i ( 1 ) ≅ H ˉ i ( 2 ) \bar H^{(1)}_i\cong\bar H^{(2)}_i H ˉ i ( 1 ) ≅ H ˉ i ( 2 ) i ∈ { 1 , … , l } i\in\{1,\ldots,l\} i ∈ { 1 , … , l } Lemma 4.20 gives us
(i) π ( G ( 1 ) , C ∅ ( G ( 1 ) ) ) = π ( G ( 2 ) , C ∅ ( G ( 2 ) ) ) , (ii) ∀ i ∈ { 1 , … , l } π ( G ( 1 ) , C { i } ( G ( 1 ) ) ) = π ( G ( 2 ) , C { i } ( G ( 2 ) ) ) . (26) \begin{aligned}
\text{(i)}\quad&\pi(G^{(1)},C_{\emptyset}(G^{(1)}))=\pi(G^{(2)},C_{\emptyset}(G^{(2)})),\\
\text{(ii)}\quad&\forall i\in\{1,\ldots,l\}\ \pi(G^{(1)},C_{\{i\}}(G^{(1)}))=\pi(G^{(2)},C_{\{i\}}(G^{(2)})). \tag*{(26)}
\end{aligned} (i) (ii) π ( G ( 1 ) , C ∅ ( G ( 1 ) )) = π ( G ( 2 ) , C ∅ ( G ( 2 ) )) , ∀ i ∈ { 1 , … , l } π ( G ( 1 ) , C { i } ( G ( 1 ) )) = π ( G ( 2 ) , C { i } ( G ( 2 ) )) . (26) Recall that C ∅ ( G ( j ) ) C_{\emptyset}(G^{(j)}) C ∅ ( G ( j ) ) H ˉ ∅ ( j ) \bar H^{(j)}_{\emptyset} H ˉ ∅ ( j ) { H ˉ i ( j ) } i ∈ { 1 , … , l } \{\bar H^{(j)}_i\}_{i\in\{1,\ldots,l\}} { H ˉ i ( j ) } i ∈ { 1 , … , l } C ∅ ( G ( j ) ) C_{\emptyset}(G^{(j)}) C ∅ ( G ( j ) ) C \mathcal C C E ( G ( j ) ) ⊆ C E(G^{(j)})\subseteq\mathcal C E ( G ( j ) ) ⊆ C C { i } ( G ( j ) ) C_{\{i\}}(G^{(j)}) C { i } ( G ( j ) ) G ( j ) G^{(j)} G ( j ) H ˉ { i } ( j ) \bar H^{(j)}_{\{i\}} H ˉ { i } ( j ) { H ˉ i ( j ) } i ∈ { 1 , … , l } \{\bar H^{(j)}_i\}_{i\in\{1,\ldots,l\}} { H ˉ i ( j ) } i ∈ { 1 , … , l } e i ( j ) e^{(j)}_i e i ( j ) e i ( j ) e^{(j)}_i e i ( j ) j = 1 , 2 j=1,2 j = 1 , 2 C ∅ ( G ( j ) ) C_{\emptyset}(G^{(j)}) C ∅ ( G ( j ) ) C { i } ( G ( j ) ) C_{\{i\}}(G^{(j)}) C { i } ( G ( j ) ) i = 1 , … , l i=1,\ldots,l i = 1 , … , l
C ∗ ( G ( j ) ) : = C ∅ ( G ( j ) ) ∪ ⋃ i = 1 l C { i } ( G ( j ) ) , j = 1 , 2 , C^*(G^{(j)}):=C_{\emptyset}(G^{(j)})\cup\bigcup_{i=1}^{l}C_{\{i\}}(G^{(j)}),\qquad j=1,2, C ∗ ( G ( j ) ) := C ∅ ( G ( j ) ) ∪ i = 1 ⋃ l C { i } ( G ( j ) ) , j = 1 , 2 , which is the family of all clique covers of G ( j ) G^{(j)} G ( j ) (26) and (8) we have
π ( G ( 1 ) , C ∗ ( G ( 1 ) ) ) = π ( G ( 2 ) , C ∗ ( G ( 2 ) ) ) . (27) \pi(G^{(1)},C^*(G^{(1)}))=\pi(G^{(2)},C^*(G^{(2)})). \tag*{(27)} π ( G ( 1 ) , C ∗ ( G ( 1 ) )) = π ( G ( 2 ) , C ∗ ( G ( 2 ) )) . (27) All clique covers that are not contained in C ∗ ( G ( j ) ) C^*(G^{(j)}) C ∗ ( G ( j ) ) k k k k ≥ 3 k\ge3 k ≥ 3 (12) we deduce
∀ C ∈ C ( G ( j ) ) ∖ C ∗ ( G ( j ) ) π ( G ( j ) , C ) ≲ m p 3 e − 2 m p 2 , j = 1 , 2. (28) \forall\mathcal C\in C(G^{(j)})\setminus C^*(G^{(j)})\ \pi(G^{(j)},\mathcal C)\lesssim mp^3e^{-2mp^2},\qquad j=1,2. \tag*{(28)} ∀ C ∈ C ( G ( j ) ) ∖ C ∗ ( G ( j ) ) π ( G ( j ) , C ) ≲ m p 3 e − 2 m p 2 , j = 1 , 2. (28) Combining (27) , (28) , and (9) we obtain
π ( H 1 ∪ H 2 ) π ( H 1 ∩ H 2 ) = π ( G ( 1 ) ) = π ( G ( 1 ) , C ∗ ( G ( 1 ) ) ) + π ( G ( 1 ) , C ( G ( 1 ) ) ∖ C ∗ ( G ( 1 ) ) ) = π ( G ( 1 ) , C ∗ ( G ( 1 ) ) ) + O ( m p 3 e − 2 m p 2 ) = π ( G ( 2 ) , C ∗ ( G ( 2 ) ) ) + O ( m p 3 e − 2 m p 2 ) = π ( G ( 2 ) , C ∗ ( G ( 2 ) ) ) + π ( G ( 2 ) , C ( G ( 2 ) ) ∖ C ∗ ( G ( 2 ) ) ) + O ( m p 3 e − 2 m p 2 ) = π ( G ( 2 ) ) + O ( m p 3 e − 2 m p 2 ) = π ( H 1 ) π ( H 2 ) + O ( m p 3 e − 2 m p 2 ) . \begin{aligned}
\pi(H_1\cup H_2)\pi(H_1\cap H_2)
&=\pi(G^{(1)})\\
&=\pi(G^{(1)},C^*(G^{(1)}))+\pi(G^{(1)},C(G^{(1)})\setminus C^*(G^{(1)}))\\
&=\pi(G^{(1)},C^*(G^{(1)}))+O(mp^3e^{-2mp^2})\\
&=\pi(G^{(2)},C^*(G^{(2)}))+O(mp^3e^{-2mp^2})\\
&=\pi(G^{(2)},C^*(G^{(2)}))+\pi(G^{(2)},C(G^{(2)})\setminus C^*(G^{(2)}))+O(mp^3e^{-2mp^2})\\
&=\pi(G^{(2)})+O(mp^3e^{-2mp^2})\\
&=\pi(H_1)\pi(H_2)+O(mp^3e^{-2mp^2}).
\end{aligned} π ( H 1 ∪ H 2 ) π ( H 1 ∩ H 2 ) = π ( G ( 1 ) ) = π ( G ( 1 ) , C ∗ ( G ( 1 ) )) + π ( G ( 1 ) , C ( G ( 1 ) ) ∖ C ∗ ( G ( 1 ) )) = π ( G ( 1 ) , C ∗ ( G ( 1 ) )) + O ( m p 3 e − 2 m p 2 ) = π ( G ( 2 ) , C ∗ ( G ( 2 ) )) + O ( m p 3 e − 2 m p 2 ) = π ( G ( 2 ) , C ∗ ( G ( 2 ) )) + π ( G ( 2 ) , C ( G ( 2 ) ) ∖ C ∗ ( G ( 2 ) )) + O ( m p 3 e − 2 m p 2 ) = π ( G ( 2 ) ) + O ( m p 3 e − 2 m p 2 ) = π ( H 1 ) π ( H 2 ) + O ( m p 3 e − 2 m p 2 ) . This, together with the fact that for m p 2 ≥ 1 mp^2\ge1 m p 2 ≥ 1 π ( H 1 ∩ H 2 ) ≍ 1 \pi(H_1\cap H_2)\asymp1 π ( H 1 ∩ H 2 ) ≍ 1
We conclude this section with a series of results that will allow us to estimate the probabilities of indecomposable clique covers.
Lemma 4.22. Let m p 2 ≤ 1 mp^2 \leq1 m p 2 ≤ 1 H 1 H_1 H 1 H 2 H_2 H 2 N j N_j N j j ≥ 0 j \geq0 j ≥ 0 j j j H = H 1 ⊕ 0 H 2 ⊕ 0 N j H = H_1 \oplus_0 H_2 \oplus_0 N_j H = H 1 ⊕ 0 H 2 ⊕ 0 N j C \mathcal{C} C { H i } i ∈ { 1 , 2 } \{H_i\}_{i\in\{1,2\}} { H i } i ∈ { 1 , 2 }
π ( H , C ) ≤ m − 1 π ( H 1 ) π ( H 2 ) . \pi(H,\mathcal{C}) \leq m^{-1}\pi(H_1)\pi(H_2). π ( H , C ) ≤ m − 1 π ( H 1 ) π ( H 2 ) . Proof. Recall that C ∈ C ( H ) \mathcal{C} \in\mathcal{C}(H) C ∈ C ( H ) { H i } i ∈ { 1 , 2 } \{H_i\}_{i\in\{1,2\}} { H i } i ∈ { 1 , 2 } i ∈ { 1 , 2 } i \in\{1,2\} i ∈ { 1 , 2 } H i H_i H i H H H Figure 8 .
Let C \mathcal{C} C H H H i = 1 , 2 i=1,2 i = 1 , 2
C i = { C ∈ C : C ⊈ V ( H i ) and C covers only edges of H i } , \mathcal{C}_i=\{C\in\mathcal{C}:C\not\subseteq V(H_i)\text{ and }C\text{ covers only edges of }H_i\}, C i = { C ∈ C : C ⊆ V ( H i ) and C covers only edges of H i } , C ∩ = { C ∈ C : C covers edges of H 1 and H 2 } . \mathcal{C}_\cap=\{C\in\mathcal{C}:C\text{ covers edges of }H_1\text{ and }H_2\}. C ∩ = { C ∈ C : C covers edges of H 1 and H 2 } . First, consider C ∈ C ∩ C\in\mathcal{C}_\cap C ∈ C ∩ H 1 H_1 H 1 H 2 H_2 H 2 ∣ C ∣ ≥ ∣ C ∩ V ( H 1 ) ∣ + ∣ C ∩ V ( H 2 ) ∣ |C|\geq|C\cap V(H_1)|+|C\cap V(H_2)| ∣ C ∣ ≥ ∣ C ∩ V ( H 1 ) ∣ + ∣ C ∩ V ( H 2 ) ∣
∀ C ∈ C ∩ m p ∣ C ∣ m p ∣ C ∩ V ( H 1 ) ∣ m p ∣ C ∩ V ( H 2 ) ∣ ≤ m − 1 . (29) \forall C\in\mathcal{C}_\cap\quad\frac{mp^{|C|}}{mp^{|C\cap V(H_1)|}mp^{|C\cap V(H_2)|}}\leq m^{-1}. \tag*{(29)} ∀ C ∈ C ∩ m p ∣ C ∩ V ( H 1 ) ∣ m p ∣ C ∩ V ( H 2 ) ∣ m p ∣ C ∣ ≤ m − 1 . (29) Now consider C ∈ C i C\in\mathcal{C}_i C ∈ C i C C C V ( H i ) V(H_i) V ( H i ) ∣ C ∣ ≥ ∣ C ∩ V ( H i ) ∣ + 1 |C|\geq|C\cap V(H_i)|+1 ∣ C ∣ ≥ ∣ C ∩ V ( H i ) ∣ + 1 m p 2 ≤ 1 mp^2\leq1 m p 2 ≤ 1
∀ C ∈ C i m p ∣ C ∣ m p ∣ C ∩ V ( H i ) ∣ ≤ p ≤ m − 1 2 . (30) \forall C\in\mathcal{C}_i\quad\frac{mp^{|C|}}{mp^{|C\cap V(H_i)|}}\leq p\leq m^{-\frac{1}{2}}. \tag*{(30)} ∀ C ∈ C i m p ∣ C ∩ V ( H i ) ∣ m p ∣ C ∣ ≤ p ≤ m − 2 1 . (30) Moreover, for any C ∈ C ∖ ( C ∩ ∪ C 1 ∪ C 2 ) C\in\mathcal{C}\setminus(\mathcal{C}_\cap\cup\mathcal{C}_1\cup\mathcal{C}_2) C ∈ C ∖ ( C ∩ ∪ C 1 ∪ C 2 ) V ( H i ) ⊃ C ∈ C [ V ( H i ) ] V(H_i)\supset C\in\mathcal{C}[V(H_i)] V ( H i ) ⊃ C ∈ C [ V ( H i )] i ∈ { 1 , 2 } i\in\{1,2\} i ∈ { 1 , 2 }
∀ C ∈ C ∖ ( C ∩ ∪ C 1 ∪ C 2 ) ∃ i ∈ { 1 , 2 } m p ∣ C ∣ = m p ∣ C ∩ V ( H i ) ∣ . (31) \forall C\in\mathcal{C}\setminus(\mathcal{C}_\cap\cup\mathcal{C}_1\cup\mathcal{C}_2)\ \exists i\in\{1,2\}\quad mp^{|C|}=mp^{|C\cap V(H_i)|}. \tag*{(31)} ∀ C ∈ C ∖ ( C ∩ ∪ C 1 ∪ C 2 ) ∃ i ∈ { 1 , 2 } m p ∣ C ∣ = m p ∣ C ∩ V ( H i ) ∣ . (31) Furthermore,
(i) for any C ∈ C ∩ C\in\mathcal{C}_\cap C ∈ C ∩ C ∩ V ( H i ) ∈ C [ V ( H i ) ] C\cap V(H_i)\in\mathcal{C}[V(H_i)] C ∩ V ( H i ) ∈ C [ V ( H i )] i = 1 , 2 i=1,2 i = 1 , 2 C ∉ C ∩ C \notin\mathcal{C}_{\cap} C ∈ / C ∩ j ∈ { 1 , 2 } j \in\{1,2\} j ∈ { 1 , 2 } C ∩ V ( H j ) ∈ C [ V ( H j ) ] C \cap V(H_j) \in\mathcal{C}[V(H_j)] C ∩ V ( H j ) ∈ C [ V ( H j )] C ∩ V ( H 3 − j ) ∉ C [ V ( H 3 − j ) ] C \cap V(H_{3-j}) \notin\mathcal{C}[V(H_{3-j})] C ∩ V ( H 3 − j ) ∈ / C [ V ( H 3 − j )] C ∈ C i C \in\mathcal{C}_i C ∈ C i j = i j=i j = i
By (10) , (31) , (i) and (ii), we get
π ( H , C ) π ( H 1 , C [ V ( H 1 ) ] ) π ( H 2 , C [ V ( H 2 ) ] ) ≲ ∏ C ∈ C m p ∣ C ∣ ∏ C ∈ C [ V ( H 1 ) ] m p ∣ C ∣ ∏ C ∈ C [ V ( H 2 ) ] m p ∣ C ∣ = ∏ C ∈ C ∩ m p ∣ C ∣ ∏ C ∈ C ∩ m p ∣ C ∩ V ( H 1 ) ∣ m p ∣ C ∩ V ( H 2 ) ∣ ⋅ ∏ C ∈ C 1 m p ∣ C ∣ ∏ C ∈ C 1 m p ∣ C ∩ V ( H 1 ) ∣ ⋅ ∏ C ∈ C 2 m p ∣ C ∣ ∏ C ∈ C 2 m p ∣ C ∩ V ( H 2 ) ∣ . \frac{\pi(H,C)}{\pi(H_1,\mathcal{C}[V(H_1)])\pi(H_2,\mathcal{C}[V(H_2)])}
\lesssim\frac{\prod_{C\in\mathcal{C}} mp^{|C|}}{\prod_{C\in\mathcal{C}[V(H_1)]}mp^{|C|}\prod_{C\in\mathcal{C}[V(H_2)]}mp^{|C|}}
= \frac{\prod_{C\in\mathcal{C}_{\cap}}mp^{|C|}}{\prod_{C\in\mathcal{C}_{\cap}}mp^{|C\cap V(H_1)|}mp^{|C\cap V(H_2)|}}\cdot\frac{\prod_{C\in\mathcal{C}_1}mp^{|C|}}{\prod_{C\in\mathcal{C}_1}mp^{|C\cap V(H_1)|}}\cdot\frac{\prod_{C\in\mathcal{C}_2}mp^{|C|}}{\prod_{C\in\mathcal{C}_2}mp^{|C\cap V(H_2)|}}. π ( H 1 , C [ V ( H 1 )]) π ( H 2 , C [ V ( H 2 )]) π ( H , C ) ≲ ∏ C ∈ C [ V ( H 1 )] m p ∣ C ∣ ∏ C ∈ C [ V ( H 2 )] m p ∣ C ∣ ∏ C ∈ C m p ∣ C ∣ = ∏ C ∈ C ∩ m p ∣ C ∩ V ( H 1 ) ∣ m p ∣ C ∩ V ( H 2 ) ∣ ∏ C ∈ C ∩ m p ∣ C ∣ ⋅ ∏ C ∈ C 1 m p ∣ C ∩ V ( H 1 ) ∣ ∏ C ∈ C 1 m p ∣ C ∣ ⋅ ∏ C ∈ C 2 m p ∣ C ∩ V ( H 2 ) ∣ ∏ C ∈ C 2 m p ∣ C ∣ . Let us note that in the last line there are no terms related to C ∈ C ∖ ( C ∩ ∪ C 1 ∪ C 2 ) C \in\mathcal{C}\setminus(\mathcal{C}_{\cap}\cup\mathcal{C}_1\cup\mathcal{C}_2) C ∈ C ∖ ( C ∩ ∪ C 1 ∪ C 2 ) (31) and (ii).
Since C \mathcal{C} C C ∈ C ∩ C\in\mathcal{C}_{\cap} C ∈ C ∩ C 1 ∈ C 1 C_1\in\mathcal{C}_1 C 1 ∈ C 1 C 2 ∈ C 2 C_2\in\mathcal{C}_2 C 2 ∈ C 2 (29) , (30) and the above equality we obtain
π ( H , C ) π ( H 1 , C [ V ( H 1 ) ] ) π ( H 2 , C [ V ( H 2 ) ] ) ≲ m − 1 . \frac{\pi(H,C)}{\pi(H_1,\mathcal{C}[V(H_1)])\pi(H_2,\mathcal{C}[V(H_2)])}\lesssim m^{-1}. π ( H 1 , C [ V ( H 1 )]) π ( H 2 , C [ V ( H 2 )]) π ( H , C ) ≲ m − 1 . This together with (8) implies the assertion.
Lemma 4.23. Let m p 2 ≥ 1 mp^2\ge1 m p 2 ≥ 1 m p 3 ≤ 1 mp^3\le1 m p 3 ≤ 1 H H H { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ C ∈ C ( H ) \mathcal{C}\in\mathcal{C}(H) C ∈ C ( H ) { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ
π ( H , C ) ≲ m p 3 e − ∣ Λ ∣ m p 2 . \pi(H,\mathcal{C})\lesssim mp^3e^{-|\Lambda|mp^2}. π ( H , C ) ≲ m p 3 e − ∣Λ∣ m p 2 . Additionally, for H = H 1 ⊕ 0 H 2 H=H_1\oplus_0 H_2 H = H 1 ⊕ 0 H 2
π ( H , C ) ≲ ( m p 3 ) 2 e − 2 m p 2 . \pi(H,\mathcal{C})\lesssim(mp^3)^2e^{-2mp^2}. π ( H , C ) ≲ ( m p 3 ) 2 e − 2 m p 2 . Proof. Each indecomposable clique cover of H H H 2 2 2 e i e_i e i i ∈ Λ i\in\Lambda i ∈ Λ H i H_i H i i ∈ Λ i\in\Lambda i ∈ Λ C i = E ( H i ) \mathcal{C}_i=E(H_i) C i = E ( H i ) i ∈ Λ i\in\Lambda i ∈ Λ e i e_i e i i ∈ Λ i\in\Lambda i ∈ Λ 2 2 2 k k k k ≥ 3 k\ge3 k ≥ 3 (12) ,
π ( H , C ) ≲ m p 3 e − ∣ Λ ∣ m p 2 , \pi(H,\mathcal{C})\lesssim mp^3e^{-|\Lambda|mp^2}, π ( H , C ) ≲ m p 3 e − ∣Λ∣ m p 2 , where the constant in the inequality may be chosen to depend only on ∣ V ( H ) ∣ |V(H)| ∣ V ( H ) ∣
In the case H = H 1 ⊕ 0 H 2 H=H_1\oplus_0 H_2 H = H 1 ⊕ 0 H 2 C \mathcal{C} C 2 2 2 e i ∈ H i e_i\in H_i e i ∈ H i i = 1 , 2 i=1,2 i = 1 , 2 H 1 H_1 H 1 H 2 H_2 H 2 e 1 e_1 e 1 e 2 e_2 e 2 4 4 4 3 3 3 (12) ,
π ( H , C ) ≲ ( ( m p 3 ) 2 + m p 4 ) e − 2 m p 2 ≲ ( m p 3 ) 2 e − 2 m p 2 . \pi(H,\mathcal{C})\lesssim\left((mp^3)^2+mp^4\right)e^{-2mp^2}\lesssim(mp^3)^2e^{-2mp^2}. π ( H , C ) ≲ ( ( m p 3 ) 2 + m p 4 ) e − 2 m p 2 ≲ ( m p 3 ) 2 e − 2 m p 2 . The proof is complete.
Definition 4.24. For a partition { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ H H H
V ∩ 1 : = { v : ∃ i ≠ j V ( H i ) ∩ V ( H j ) = { v } } , V_{\cap1}:=\{v:\exists_{i\ne j}V(H_i)\cap V(H_j)=\{v\}\}, V ∩ 1 := { v : ∃ i = j V ( H i ) ∩ V ( H j ) = { v }} , i.e. the set of all the vertices that belong to at least two graphs from the partition. Furthermore, define the bipartite graph F = F ( H , ( H i ) i ∈ Λ ) F=F(H,(H_i)_{i\in\Lambda}) F = F ( H , ( H i ) i ∈ Λ ) ( Λ , V ∩ 1 ) (\Lambda,V_{\cap1}) ( Λ , V ∩ 1 )
V ( F ) : = Λ ∪ V ∩ 1 , V(F):=\Lambda\cup V_{\cap1}, V ( F ) := Λ ∪ V ∩ 1 , E ( F ) : = { { i , v } : i ∈ Λ , v ∈ V ∩ 1 , v ∈ V ( H i ) } . E(F):=\{\{i,v\}:i\in\Lambda,v\in V_{\cap1},v\in V(H_i)\}. E ( F ) := {{ i , v } : i ∈ Λ , v ∈ V ∩ 1 , v ∈ V ( H i )} . In other words, { i , v } ∈ E ( F ) \{i,v\}\in E(F) { i , v } ∈ E ( F ) v ∈ V ( H i ∩ H j ) v\in V(H_i\cap H_j) v ∈ V ( H i ∩ H j ) j ∈ Λ j\in\Lambda j ∈ Λ F = F ( H , ( H i ) i ∈ Λ ) F=F(H,(H_i)_{i\in\Lambda}) F = F ( H , ( H i ) i ∈ Λ ) Figure 9 .
Figure 9. Graphs F ( G t , ( K a b ) a b ∈ Γ ) F(G_{t},(K_{ab})_{ab\in\Gamma}) F ( G t , ( K ab ) ab ∈ Γ ) t = 1 , 2 , 3 t=1,2,3 t = 1 , 2 , 3 Λ = Γ = { 11 , 12 , 21 , 22 } \Lambda=\Gamma=\{11,12,21,22\} Λ = Γ = { 11 , 12 , 21 , 22 } H a b = K a b H_{ab}=K_{ab} H ab = K ab a b ∈ Λ ab\in\Lambda ab ∈ Λ V ∩ 1 V_{\cap1} V ∩ 1 Λ = Γ \Lambda=\Gamma Λ = Γ
Lemma 4.25. Let m p ≥ 1 m_p\ge1 m p ≥ 1 m p 2 ≤ 1 m_p^2\le1 m p 2 ≤ 1 H H H ( H i ) i ∈ Λ (H_i)_{i\in\Lambda} ( H i ) i ∈ Λ F = F ( H , { H i } i ∈ Λ ) F=F(H,\{H_i\}_{i\in\Lambda}) F = F ( H , { H i } i ∈ Λ ) C ∈ C ( H ) C\in\mathcal{C}(H) C ∈ C ( H ) { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ H H H
π ( H , C ) ≲ ∏ i ∈ Λ π ( H i ) ( m p ) ⌈ ∣ Λ ∣ / 2 ⌉ p δ , \pi(H,C)\lesssim\frac{\prod_{i\in\Lambda}\pi(H_i)}{(m_p)^{\lceil|\Lambda|/2\rceil}p^\delta}, π ( H , C ) ≲ ( m p ) ⌈ ∣Λ∣/2 ⌉ p δ ∏ i ∈ Λ π ( H i ) , where δ = 1 \delta=1 δ = 1 F F F δ = 0 \delta=0 δ = 0
Proof. Let C ∈ C ( H ) C\in\mathcal{C}(H) C ∈ C ( H ) { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ i ∈ Λ i\in\Lambda i ∈ Λ e i ∈ E ( H i ) e_i\in E(H_i) e i ∈ E ( H i ) V ( H i ) V(H_i) V ( H i ) C C C P1 then by Lemma 4.7 (ii), it may be replaced by C ′ C' C ′ P1 such that each e i e_i e i i ∈ Λ i\in\Lambda i ∈ Λ C i C_i C i P1 ). Moreover, Lemma 4.7 (ii) implies that C i ⊈ V ( H i ) C_i\nsubseteq V(H_i) C i ⊈ V ( H i ) e i e_i e i C C C V ( H i ) V(H_i) V ( H i ) C ′ C' C ′ C C C P1 .
Let C ∗ = { C 1 , … , C s } ⊆ C C^*=\{C_1,\ldots,C_s\}\subseteq C C ∗ = { C 1 , … , C s } ⊆ C C ∈ C C\in C C ∈ C
∀ i ∈ Λ C ⊈ V ( H i ) , \forall_{i\in\Lambda}C\nsubseteq V(H_i), ∀ i ∈ Λ C ⊈ V ( H i ) , i.e., sets that are not contained in the vertex set of any of the graphs of the partition.
For all j ∈ { 1 , … , s } j \in\{1,\ldots,s\} j ∈ { 1 , … , s }
Λ j : = { i ∈ Λ : C j covers an edge of H i } , \Lambda_j:=\{i\in\Lambda:C_j\text{ covers an edge of }H_i\}, Λ j := { i ∈ Λ : C j covers an edge of H i } , that is the index set of graphs from the partition whose edges are covered by C j C_j C j C \mathcal{C} C i ∈ Λ i\in\Lambda i ∈ Λ H i H_i H i C ∈ C ∗ C\in\mathcal{C}^* C ∈ C ∗ ⋃ j = 1 s Λ j = Λ \bigcup_{j=1}^{s}\Lambda_j=\Lambda ⋃ j = 1 s Λ j = Λ
∑ j = 1 s ∣ Λ j ∣ ≥ ∣ Λ ∣ . (32) \sum_{j=1}^{s}|\Lambda_j|\geq|\Lambda|. \tag*{(32)} j = 1 ∑ s ∣ Λ j ∣ ≥ ∣Λ∣. (32) Now we define a clique cover
C H = ⋃ i ∈ Λ C [ V ( H i ) ] . (33) \mathcal{C}_H=\bigcup_{i\in\Lambda}\mathcal{C}[V(H_i)]. \tag*{(33)} C H = i ∈ Λ ⋃ C [ V ( H i )] . (33) Note that
∀ C ∉ C ∗ ∃ i ∈ Λ C ⊆ V ( H i ) and m p ∣ C ∣ = m p ∣ C ∩ V ( H i ) ∣ . \forall C\notin\mathcal{C}^*\ \exists i\in\Lambda_C\subseteq V(H_i)\text{ and }mp^{|C|}=mp^{|C\cap V(H_i)|}. ∀ C ∈ / C ∗ ∃ i ∈ Λ C ⊆ V ( H i ) and m p ∣ C ∣ = m p ∣ C ∩ V ( H i ) ∣ . Hence, by (10) , since the terms related to C ∉ C ∗ C\notin\mathcal{C}^* C ∈ / C ∗
π ( H , C ) π ( H , C H ) ≲ ∏ C ∈ C m p ∣ C ∣ ∏ i ∈ Λ ∏ C ′ ∈ C [ H i ] m p ∣ C ′ ∣ ≲ ∏ j = 1 s m p ∣ C j ∣ ∏ i ∈ Λ j m p ∣ C j ∩ V ( H i ) ∣ = ∏ j = 1 s m − ( ∣ Λ j ∣ − 1 ) p ∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ H i ∣ . (34) \frac{\pi(H,\mathcal{C})}{\pi(H,\mathcal{C}_H)}
\lesssim\frac{\prod_{C\in\mathcal{C}}mp^{|C|}}{\prod_{i\in\Lambda}\prod_{C'\in\mathcal{C}[H_i]}mp^{|C'|}}
\lesssim\prod_{j=1}^{s}\frac{mp^{|C_j|}}{\prod_{i\in\Lambda_j}mp^{|C_j\cap V(H_i)|}}
=\prod_{j=1}^{s}m^{-(|\Lambda_j|-1)p^{|C_j|-\sum_{i\in\Lambda_j}|C_j\cap H_i|}}. \tag*{(34)} π ( H , C H ) π ( H , C ) ≲ ∏ i ∈ Λ ∏ C ′ ∈ C [ H i ] m p ∣ C ′ ∣ ∏ C ∈ C m p ∣ C ∣ ≲ j = 1 ∏ s ∏ i ∈ Λ j m p ∣ C j ∩ V ( H i ) ∣ m p ∣ C j ∣ = j = 1 ∏ s m − ( ∣ Λ j ∣ − 1 ) p ∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ H i ∣ . (34) Now we analyse each term of the product.
If ∣ Λ j ∣ = 1 |\Lambda_j|=1 ∣ Λ j ∣ = 1 l ∈ Λ l\in\Lambda l ∈ Λ Λ j = { l } \Lambda_j=\{l\} Λ j = { l } ∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ V ( H i ) ∣ = ∣ C j ∣ − ∣ C j ∩ V ( H l ) ∣ ≥ 1 |C_j|-\sum_{i\in\Lambda_j}|C_j\cap V(H_i)|=|C_j|-|C_j\cap V(H_l)|\geq1 ∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ V ( H i ) ∣ = ∣ C j ∣ − ∣ C j ∩ V ( H l ) ∣ ≥ 1 C j ⊈ V ( H l ) C_j\not\subseteq V(H_l) C j ⊆ V ( H l ) m p 2 ≤ 1 mp^2\leq1 m p 2 ≤ 1
m − ( ∣ Λ j ∣ − 1 ) p ∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ H i ∣ ≤ p ≤ 1 m p . (35) m^{-(|\Lambda_j|-1)p^{|C_j|-\sum_{i\in\Lambda_j}|C_j\cap H_i|}}\leq p\leq\frac{1}{mp}. \tag*{(35)} m − ( ∣ Λ j ∣ − 1 ) p ∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ H i ∣ ≤ p ≤ m p 1 . (35) Now consider ∣ Λ j ∣ ≥ 2 |\Lambda_j|\geq2 ∣ Λ j ∣ ≥ 2 V ∩ 1 V_{\cap1} V ∩ 1 H H H { H i } i ∈ Λ \{H_i\}_{i\in\Lambda} { H i } i ∈ Λ V ( H i ) ∖ V ∩ 1 V(H_i)\setminus V_{\cap1} V ( H i ) ∖ V ∩ 1 i ∈ Λ i\in\Lambda i ∈ Λ j ∈ { 1 , … , s } j\in\{1,\ldots,s\} j ∈ { 1 , … , s } ⋃ i ∈ Λ j ( V ( H i ) ∖ V ∩ 1 ) ⊆ ( V ( H ) ∖ V ∩ 1 ) \bigcup_{i\in\Lambda_j}(V(H_i)\setminus V_{\cap1})\subseteq(V(H)\setminus V_{\cap1}) ⋃ i ∈ Λ j ( V ( H i ) ∖ V ∩ 1 ) ⊆ ( V ( H ) ∖ V ∩ 1 )
∑ i ∈ Λ j ∣ ( C j ∩ ( V ( H i ) ∖ V ∩ 1 ) ) ∣ = ∣ ( C j ∩ ⋃ i ∈ Λ j V ( H i ) ) ∖ V ∩ 1 ∣ ≤ ∣ C j ∖ V ∩ 1 ∣ . \sum_{i\in\Lambda_j}\left|(C_j\cap(V(H_i)\setminus V_{\cap1}))\right|
=\left|\left(C_j\cap\bigcup_{i\in\Lambda_j}V(H_i)\right)\setminus V_{\cap1}\right|
\leq|C_j\setminus V_{\cap1}|. i ∈ Λ j ∑ ∣ ( C j ∩ ( V ( H i ) ∖ V ∩ 1 )) ∣ = C j ∩ i ∈ Λ j ⋃ V ( H i ) ∖ V ∩ 1 ≤ ∣ C j ∖ V ∩ 1 ∣. Thus
∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ V ( H i ) ∣ = ∣ C j ∖ V ∩ 1 ∣ + ∣ C j ∩ V ∩ 1 ∣ − ∑ i ∈ Λ j ( ∣ ( C j ∩ V ( H i ) ) ∖ V ∩ 1 ∣ + ∣ C j ∩ V ( H i ) ∩ V ∩ 1 ∣ ) ≥ ∣ C j ∩ V ∩ 1 ∣ − ∑ i ∈ Λ j ∣ C j ∩ V ( H i ) ∩ V ∩ 1 ∣ . = ∣ C j ∩ V ∩ 1 ∣ − ∑ i ∈ Λ j ∑ v ∈ V ∩ 1 ∩ C j ∣ V ( H i ) ∩ { v } ∣ = ∣ C j ∩ V ∩ 1 ∣ − ∑ v ∈ V ∩ 1 ∩ C j ∑ i ∈ Λ j ∣ V ( H i ) ∩ { v } ∣ . (36) \begin{aligned}
|C_j|-\sum_{i\in\Lambda_j}|C_j\cap V(H_i)|
&=|C_j\setminus V_{\cap1}|+|C_j\cap V_{\cap1}|\\
&\quad-\sum_{i\in\Lambda_j}\left(|(C_j\cap V(H_i))\setminus V_{\cap1}|+|C_j\cap V(H_i)\cap V_{\cap1}|\right)\\
&\geq|C_j\cap V_{\cap1}|-\sum_{i\in\Lambda_j}|C_j\cap V(H_i)\cap V_{\cap1}|. \\
&= |C_j \cap V_{\cap1}| - \sum_{i\in\Lambda_j}\sum_{v\in V_{\cap1}\cap C_j} |V(H_i)\cap\{v\}|
= |C_j \cap V_{\cap1}| - \sum_{v\in V_{\cap1}\cap C_j}\sum_{i\in\Lambda_j} |V(H_i)\cap\{v\}|. \tag*{(36)}
\end{aligned} ∣ C j ∣ − i ∈ Λ j ∑ ∣ C j ∩ V ( H i ) ∣ = ∣ C j ∖ V ∩ 1 ∣ + ∣ C j ∩ V ∩ 1 ∣ − i ∈ Λ j ∑ ( ∣ ( C j ∩ V ( H i )) ∖ V ∩ 1 ∣ + ∣ C j ∩ V ( H i ) ∩ V ∩ 1 ∣ ) ≥ ∣ C j ∩ V ∩ 1 ∣ − i ∈ Λ j ∑ ∣ C j ∩ V ( H i ) ∩ V ∩ 1 ∣. = ∣ C j ∩ V ∩ 1 ∣ − i ∈ Λ j ∑ v ∈ V ∩ 1 ∩ C j ∑ ∣ V ( H i ) ∩ { v } ∣ = ∣ C j ∩ V ∩ 1 ∣ − v ∈ V ∩ 1 ∩ C j ∑ i ∈ Λ j ∑ ∣ V ( H i ) ∩ { v } ∣. (36) Recall that the degree of v ∈ V ∩ 1 v\in V_{\cap1} v ∈ V ∩ 1 F F F H i H_i H i i ∈ Λ i\in\Lambda i ∈ Λ v v v ∑ i ∈ Λ j ∣ V ( H i ) ∩ { v } ∣ \sum_{i\in\Lambda_j}|V(H_i)\cap\{v\}| ∑ i ∈ Λ j ∣ V ( H i ) ∩ { v } ∣ H i H_i H i i ∈ Λ j i\in\Lambda_j i ∈ Λ j v v v v v v F F F Λ j ∪ ( V ∩ 1 ∩ C j ) \Lambda_j\cup(V_{\cap1}\cap C_j) Λ j ∪ ( V ∩ 1 ∩ C j ) v ∈ C j ∩ V ∩ 1 v\in C_j\cap V_{\cap1} v ∈ C j ∩ V ∩ 1
∑ i ∈ Λ j ∣ V ( H i ) ∩ { v } ∣ = deg F j ( v ) , where F j = F [ Λ j ∪ ( V ∩ 1 ∩ C j ) ] . \sum_{i\in\Lambda_j}|V(H_i)\cap\{v\}|=\deg_{F_j}(v),\qquad\text{where }F_j=F[\Lambda_j\cup(V_{\cap1}\cap C_j)]. i ∈ Λ j ∑ ∣ V ( H i ) ∩ { v } ∣ = deg F j ( v ) , where F j = F [ Λ j ∪ ( V ∩ 1 ∩ C j )] . Moreover, since F j F_j F j ( V ∩ 1 ∩ C j , Λ j ) (V_{\cap1}\cap C_j,\Lambda_j) ( V ∩ 1 ∩ C j , Λ j )
∑ v ∈ V ∩ 1 ∩ C j ∑ i ∈ Λ j ∣ V ( H i ) ∩ { v } ∣ = ∑ v ∈ V ∩ 1 ∩ C j deg F j ( v ) = ∣ E ( F j ) ∣ . \sum_{v\in V_{\cap1}\cap C_j}\sum_{i\in\Lambda_j}|V(H_i)\cap\{v\}|=\sum_{v\in V_{\cap1}\cap C_j}\deg_{F_j}(v)=|E(F_j)|. v ∈ V ∩ 1 ∩ C j ∑ i ∈ Λ j ∑ ∣ V ( H i ) ∩ { v } ∣ = v ∈ V ∩ 1 ∩ C j ∑ deg F j ( v ) = ∣ E ( F j ) ∣. Therefore, when we substitute this equation to (36) we obtain
∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ V ( H i ) ∣ = ∣ V ∩ 1 ∩ C j ∣ − ∣ E ( F j ) ∣ . |C_j|-\sum_{i\in\Lambda_j}|C_j\cap V(H_i)|=|V_{\cap1}\cap C_j|-|E(F_j)|. ∣ C j ∣ − i ∈ Λ j ∑ ∣ C j ∩ V ( H i ) ∣ = ∣ V ∩ 1 ∩ C j ∣ − ∣ E ( F j ) ∣. Let us define δ j = 1 \delta_j=1 δ j = 1 F j F_j F j δ j = 0 \delta_j=0 δ j = 0 F j F_j F j F F F F j F_j F j j = 1 , … , s j=1,\ldots,s j = 1 , … , s
∣ E ( F j ) ∣ ≤ ∣ V ( F j ) ∣ − 1 + δ j = ∣ Λ j ∣ + ∣ C j ∩ V ∩ 1 ∣ − 1 + δ j . |E(F_j)|\leq|V(F_j)|-1+\delta_j=|\Lambda_j|+|C_j\cap V_{\cap1}|-1+\delta_j. ∣ E ( F j ) ∣ ≤ ∣ V ( F j ) ∣ − 1 + δ j = ∣ Λ j ∣ + ∣ C j ∩ V ∩ 1 ∣ − 1 + δ j . Combining the last two equalities
∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ V ( H i ) ∣ ≥ − ∣ Λ j ∣ + 1 − δ j . |C_j|-\sum_{i\in\Lambda_j}|C_j\cap V(H_i)|\geq-|\Lambda_j|+1-\delta_j. ∣ C j ∣ − i ∈ Λ j ∑ ∣ C j ∩ V ( H i ) ∣ ≥ − ∣ Λ j ∣ + 1 − δ j . Therefore
m − ( ∣ Λ j ∣ − 1 ) p ∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ H i ∣ ≤ m − ( ∣ Λ j ∣ − 1 ) p − ∣ Λ j ∣ + 1 − δ j = ( m p ) − ( ∣ Λ j ∣ − 1 ) p − δ j . m^{-(|\Lambda_j|-1)}p^{|C_j|-\sum_{i\in\Lambda_j}|C_j\cap H_i|}
\leq m^{-(|\Lambda_j|-1)}p^{-|\Lambda_j|+1-\delta_j}
=(mp)^{-(|\Lambda_j|-1)}p^{-\delta_j}. m − ( ∣ Λ j ∣ − 1 ) p ∣ C j ∣ − ∑ i ∈ Λ j ∣ C j ∩ H i ∣ ≤ m − ( ∣ Λ j ∣ − 1 ) p − ∣ Λ j ∣ + 1 − δ j = ( m p ) − ( ∣ Λ j ∣ − 1 ) p − δ j . Substituting this and (35) to (34) we obtain
π ( H , C ) π ( H , C H ) ≲ ( ∏ j , ∣ Λ j ∣ = 1 ( m p ) − 1 ) ⋅ ( ∏ j , ∣ Λ j ∣ ≥ 2 m − ( ∣ Λ j ∣ − 1 ) p − ∣ Λ j ∣ + 1 − δ j ) = 1 ( m p ) ∑ j , ∣ Λ j ∣ = 1 ∣ Λ j ∣ + ∑ j , ∣ Λ j ∣ ≥ 2 ( ∣ Λ j ∣ − 1 ) p ∑ j = 1 s δ j . (37) \frac{\pi(H,C)}{\pi(H,C_H)}
\lesssim
\left(\prod_{j,|\Lambda_j|=1}(mp)^{-1}\right)\cdot
\left(\prod_{j,|\Lambda_j|\geq2}m^{-(|\Lambda_j|-1)}p^{-|\Lambda_j|+1-\delta_j}\right)
=
\frac{1}{(mp)^{\sum_{j,|\Lambda_j|=1}|\Lambda_j|+\sum_{j,|\Lambda_j|\geq2}(|\Lambda_j|-1)}p^{\sum_{j=1}^{s}\delta_j}}. \tag*{(37)} π ( H , C H ) π ( H , C ) ≲ j , ∣ Λ j ∣ = 1 ∏ ( m p ) − 1 ⋅ j , ∣ Λ j ∣ ≥ 2 ∏ m − ( ∣ Λ j ∣ − 1 ) p − ∣ Λ j ∣ + 1 − δ j = ( m p ) ∑ j , ∣ Λ j ∣ = 1 ∣ Λ j ∣ + ∑ j , ∣ Λ j ∣ ≥ 2 ( ∣ Λ j ∣ − 1 ) p ∑ j = 1 s δ j 1 . (37) Since C C C P1 , the inequality ∣ C j 1 ∩ C j 2 ∣ ≤ 1 |C_{j_1}\cap C_{j_2}|\leq1 ∣ C j 1 ∩ C j 2 ∣ ≤ 1 j 1 , j 2 ∈ { 1 , … , s } j_1,j_2\in\{1,\ldots,s\} j 1 , j 2 ∈ { 1 , … , s } j 1 ≠ j 2 j_1\neq j_2 j 1 = j 2 F j 1 F_{j_1} F j 1 F j 2 F_{j_2} F j 2 V ∩ 1 V_{\cap1} V ∩ 1 F F F F j F_j F j j = 1 , … , s j=1,\ldots,s j = 1 , … , s F F F V η 1 V_{\eta_1} V η 1 δ j = 1 \delta_j=1 δ j = 1 j j j j = 1 , … , s j=1,\ldots,s j = 1 , … , s F F F
∑ j = 1 s δ j ≤ δ . \sum_{j=1}^{s}\delta_j\leq\delta. j = 1 ∑ s δ j ≤ δ . Moreover, using (32) ,
∑ j , ∣ Λ j ∣ = 1 ∣ Λ j ∣ + ∑ j , ∣ Λ j ∣ ≥ 2 ( ∣ Λ j ∣ − 1 ) ≥ ⌈ ∑ j ∣ Λ j ∣ 2 ⌉ ≥ ⌈ ∣ Λ ∣ 2 ⌉ . \sum_{j,|\Lambda_j|=1}|\Lambda_j|+\sum_{j,|\Lambda_j|\geq2}(|\Lambda_j|-1)\geq\left\lceil\frac{\sum_j|\Lambda_j|}{2}\right\rceil\geq\left\lceil\frac{|\Lambda|}{2}\right\rceil. j , ∣ Λ j ∣ = 1 ∑ ∣ Λ j ∣ + j , ∣ Λ j ∣ ≥ 2 ∑ ( ∣ Λ j ∣ − 1 ) ≥ ⌈ 2 ∑ j ∣ Λ j ∣ ⌉ ≥ ⌈ 2 ∣Λ∣ ⌉ . The two equalities above together with (37) imply
π ( H , C ) π ( H , C H ) ≲ 1 ( m p ) ⌈ ∣ Λ ∣ / 2 ⌉ p δ . \frac{\pi(H,\mathcal{C})}{\pi(H,\mathcal{C}_H)}\lesssim\frac{1}{(mp)^{\lceil|\Lambda|/2\rceil}p^\delta}. π ( H , C H ) π ( H , C ) ≲ ( m p ) ⌈ ∣Λ∣/2 ⌉ p δ 1 . Thus, recalling the definition of C H \mathcal{C}_H C H (33) , by (10) and the fact that graphs H i H_i H i i ∈ Λ i\in\Lambda i ∈ Λ C [ V ( H i ) ] \mathcal{C}[V(H_i)] C [ V ( H i )] i ∈ Λ i\in\Lambda i ∈ Λ
π ( H , C ) ≲ π ( H , ⋃ i ∈ Λ C [ V ( H i ) ] ) ( m p ) ⌈ ∣ Λ ∣ / 2 ⌉ p δ ≲ ∏ i ∈ Λ π ( H i , C [ V ( H i ) ] ) ( m p ) ⌈ ∣ Λ ∣ / 2 ⌉ p δ ≲ ∏ i ∈ Λ π ( H i ) ( m p ) ⌈ ∣ Λ ∣ / 2 ⌉ p δ . \pi(H,\mathcal{C})\lesssim\frac{\pi\left(H,\bigcup_{i\in\Lambda}\mathcal{C}[V(H_i)]\right)}{(mp)^{\lceil|\Lambda|/2\rceil}p^\delta}\lesssim\frac{\prod_{i\in\Lambda}\pi\left(H_i,\mathcal{C}[V(H_i)]\right)}{(mp)^{\lceil|\Lambda|/2\rceil}p^\delta}\lesssim\frac{\prod_{i\in\Lambda}\pi(H_i)}{(mp)^{\lceil|\Lambda|/2\rceil}p^\delta}. π ( H , C ) ≲ ( m p ) ⌈ ∣Λ∣/2 ⌉ p δ π ( H , ⋃ i ∈ Λ C [ V ( H i )] ) ≲ ( m p ) ⌈ ∣Λ∣/2 ⌉ p δ ∏ i ∈ Λ π ( H i , C [ V ( H i )] ) ≲ ( m p ) ⌈ ∣Λ∣/2 ⌉ p δ ∏ i ∈ Λ π ( H i ) . The proof is complete.
Proof of Theorem 2.1
Recall that K \mathcal{K} K r r r V V V
X n ( r ) = ∑ K ∈ K I K . X_n^{(r)}=\sum_{K\in\mathcal{K}}\mathbb{I}_K. X n ( r ) = K ∈ K ∑ I K . where I K \mathbb{I}_K I K { K ⊆ G ( n , m , p ) } \{K\subseteq G(n,m,p)\} { K ⊆ G ( n , m , p )}
In this section we study the variance of X n ( r ) X_n^{(r)} X n ( r ) I K \mathbb{I}_K I K I K ′ \mathbb{I}_{K'} I K ′ K , K ′ K,K' K , K ′
Var [ X n ( r ) ] = ∑ K ∈ K Var [ I K ] + ∑ K 1 , K 2 ∈ K K 1 ≠ K 2 Cov ( I K 1 , I K 2 ) = ∑ K ∈ K Var [ I K ] + ∑ i = 1 r − 1 ∑ K 1 , K 2 ∈ K ∣ V ( K 1 ∩ K 2 ) ∣ = i Cov ( I K 1 , I K 2 ) = ( n r ) P ( 1 − P ) + ∑ i = 1 r − 1 ∣ { K 1 , K 2 ∈ K : ∣ V ( K 1 ∩ K 2 ) ∣ = i } ∣ ( π ( K 1 ⊕ i K 2 ) − P 2 ) = ∑ i = 1 r ( n ) 2 r − i [ ( r − i ) ! ] 2 i ! ( π ( K 1 ⊕ i K 2 ) − P 2 ) , (38) \begin{aligned}
\operatorname{Var}[X_n^{(r)}]
&=\sum_{K\in\mathcal{K}}\operatorname{Var}[\mathbb{I}_K]
+\sum_{\substack{K_1,K_2\in\mathcal{K}\\K_1\ne K_2}}\operatorname{Cov}(\mathbb{I}_{K_1},\mathbb{I}_{K_2})\\
&=\sum_{K\in\mathcal{K}}\operatorname{Var}[\mathbb{I}_K]
+\sum_{i=1}^{r-1}\sum_{\substack{K_1,K_2\in\mathcal{K}\\|V(K_1\cap K_2)|=i}}\operatorname{Cov}(\mathbb{I}_{K_1},\mathbb{I}_{K_2})\\
&=\binom{n}{r}\mathsf{P}(1-\mathsf{P})
+\sum_{i=1}^{r-1}\left|\{K_1,K_2\in\mathcal{K}:|V(K_1\cap K_2)|=i\}\right|\left(\pi(K_1\oplus_i K_2)-\mathsf{P}^2\right)\\
&=\sum_{i=1}^{r}\frac{(n)_{2r-i}}{[(r-i)!]^2i!}\left(\pi(K_1\oplus_i K_2)-\mathsf{P}^2\right), \tag*{(38)}
\end{aligned} Var [ X n ( r ) ] = K ∈ K ∑ Var [ I K ] + K 1 , K 2 ∈ K K 1 = K 2 ∑ Cov ( I K 1 , I K 2 ) = K ∈ K ∑ Var [ I K ] + i = 1 ∑ r − 1 K 1 , K 2 ∈ K ∣ V ( K 1 ∩ K 2 ) ∣ = i ∑ Cov ( I K 1 , I K 2 ) = ( r n ) P ( 1 − P ) + i = 1 ∑ r − 1 ∣ { K 1 , K 2 ∈ K : ∣ V ( K 1 ∩ K 2 ) ∣ = i } ∣ ( π ( K 1 ⊕ i K 2 ) − P 2 ) = i = 1 ∑ r [( r − i )! ] 2 i ! ( n ) 2 r − i ( π ( K 1 ⊕ i K 2 ) − P 2 ) , (38) where ( n ) k = n ⋅ ( n − 1 ) ⋅ ( n − k + 1 ) (n)_k=n\cdot(n-1)\cdot(n-k+1) ( n ) k = n ⋅ ( n − 1 ) ⋅ ( n − k + 1 ) K 1 , K 2 K_1,K_2 K 1 , K 2 r r r (38) are non-negative.
In the proofs of this section, we will use the following relations concerning the terms in (3) , that are consequences of (1) and Fact 4.2
( n r P ) 2 Ψ ( 1 − p ) ≲ n 2 r − 2 e − m p 2 , for m p 2 ≥ 1 , \frac{(n^r\mathbf{P})^2}{\Psi}(1-\mathbf{p}) \lesssim n^{2r-2}e^{-mp^2}, \qquad\text{for } mp^2 \ge1, Ψ ( n r P ) 2 ( 1 − p ) ≲ n 2 r − 2 e − m p 2 , for m p 2 ≥ 1 , ( P p ) 2 m p 3 ( 1 − p ) 2 ≲ { P 2 m p , for m p 2 ≤ 1 ; m p 3 e − 2 m p 2 , for m p 2 ≥ 1 and m p 3 ≤ 1. (39) \left(\frac{\mathbf{P}}{p}\right)^2 mp^3(1-p)^2 \lesssim
\begin{cases}
\frac{\mathbf{P}^2}{mp}, & \text{for } mp^2 \le1;\\
mp^3e^{-2mp^2}, & \text{for } mp^2 \ge1 \text{ and } mp^3 \le1.
\end{cases} \tag*{(39)} ( p P ) 2 m p 3 ( 1 − p ) 2 ≲ { m p P 2 , m p 3 e − 2 m p 2 , for m p 2 ≤ 1 ; for m p 2 ≥ 1 and m p 3 ≤ 1. (39) In particular, this allows us to rewrite (3) into the form
Var [ X n ( r ) ] ≲ { ( n r P ) 2 Ψ Λ ( n m p ) , for m p 2 ≤ 1 ; n 2 r − 2 e − m p 2 + n 2 r − 1 m p 3 e − 2 m p 2 , for m p 2 ≥ 1 and m p 3 ≤ c r . (40) \operatorname{Var}[X_n^{(r)}] \lesssim
\begin{cases}
\frac{(n^r\mathbf{P})^2}{\Psi\Lambda(nmp)}, & \text{for } mp^2 \le1;\\
n^{2r-2}e^{-mp^2}+n^{2r-1}mp^3e^{-2mp^2}, & \text{for } mp^2 \ge1 \text{ and } mp^3 \le c_r.
\end{cases} \tag*{(40)} Var [ X n ( r ) ] ≲ { ΨΛ ( nm p ) ( n r P ) 2 , n 2 r − 2 e − m p 2 + n 2 r − 1 m p 3 e − 2 m p 2 , for m p 2 ≤ 1 ; for m p 2 ≥ 1 and m p 3 ≤ c r . (40) Let us recall the definition (14) of Ψ i \Psi_i Ψ i (38) , the next two lemmas, and the equivalence (15) .
Lemma 5.1. Let m m m 2 ≤ i ≤ r 2 \le i \le r 2 ≤ i ≤ r m p ≤ 1 mp \le1 m p ≤ 1 m p 2 ≥ 1 mp^2 \ge1 m p 2 ≥ 1 m p 3 ≤ 1 mp^3 \le1 m p 3 ≤ 1
n 2 r − i π ( K 1 ⊕ i K 2 ) ≍ ( n r P ) 2 Ψ i , (41) n^{2r-i}\pi(\mathrm{K}_1\oplus_i\mathrm{K}_2) \asymp\frac{(n^r\mathbf{P})^2}{\Psi_i}, \tag*{(41)} n 2 r − i π ( K 1 ⊕ i K 2 ) ≍ Ψ i ( n r P ) 2 , (41) while if m p ≥ 1 mp \ge1 m p ≥ 1 m p 2 ≤ 1 mp^2 \le1 m p 2 ≤ 1
( n r P ) 2 Ψ i ≲ n 2 r − i π ( K 1 ⊕ i K 2 ) ≲ ( n r P ) 2 Ψ . (42) \frac{(n^r\mathbf{P})^2}{\Psi_i} \lesssim n^{2r-i}\pi(\mathrm{K}_1\oplus_i\mathrm{K}_2) \lesssim\frac{(n^r\mathbf{P})^2}{\Psi}. \tag*{(42)} Ψ i ( n r P ) 2 ≲ n 2 r − i π ( K 1 ⊕ i K 2 ) ≲ Ψ ( n r P ) 2 . (42) Furthermore, for m p 3 ≤ 1 mp^3 \le1 m p 3 ≤ 1
π ( K 1 ⊕ i K 2 ) − P 2 ≍ ( 1 − p ) π ( K 1 ⊕ i K 2 ) . (43) \pi(\mathrm{K}_1\oplus_i\mathrm{K}_2)-\mathbf{P}^2 \asymp(1-p)\pi(\mathrm{K}_1\oplus_i\mathrm{K}_2). \tag*{(43)} π ( K 1 ⊕ i K 2 ) − P 2 ≍ ( 1 − p ) π ( K 1 ⊕ i K 2 ) . (43) Note that in (42) the lower bound may not be of the same order as the upper one. This follows from the fact that, for certain parameter ranges, we have π ( K ) ≍ π ( K , E ( K ) ) \pi(\mathrm{K})\asymp\pi(\mathrm{K},E(\mathrm{K})) π ( K ) ≍ π ( K , E ( K )) π ( K ⊕ i K ) ≍ π ( K ⊕ i K , C ) \pi(\mathrm{K}\oplus_i\mathrm{K})\asymp\pi(\mathrm{K}\oplus_i\mathrm{K},\mathcal{C}) π ( K ⊕ i K ) ≍ π ( K ⊕ i K , C ) C \mathcal{C} C C = { V ( K ⊕ i K ) } \mathcal{C}=\{V(\mathrm{K}\oplus_i\mathrm{K})\} C = { V ( K ⊕ i K )}
Proof. Assume first m p ≤ 1 mp\le1 m p ≤ 1 (10) and Lemma 4.5, for any 2 ≤ i ≤ r 2\le i\le r 2 ≤ i ≤ r
n 2 r − i π ( K 1 ⊕ i K 2 ) ≍ n 2 r − i π ( K 1 ⊕ i K 2 , { V ( K 1 ⊕ i K 2 ) } ) ≍ n 2 r − i m p 2 r − i ≍ ( n r P ) 2 Ψ i , n^{2r-i}\pi(\mathrm{K}_1\oplus_i\mathrm{K}_2)\asymp n^{2r-i}\pi(\mathrm{K}_1\oplus_i\mathrm{K}_2,\{V(\mathrm{K}_1\oplus_i\mathrm{K}_2)\})\asymp n^{2r-i}mp^{2r-i}\asymp\frac{(n^r\mathbf{P})^2}{\Psi_i}, n 2 r − i π ( K 1 ⊕ i K 2 ) ≍ n 2 r − i π ( K 1 ⊕ i K 2 , { V ( K 1 ⊕ i K 2 )}) ≍ n 2 r − i m p 2 r − i ≍ Ψ i ( n r P ) 2 , which implies (41) . Using Corollary 4.1 once again we obtain
P 2 π ( K 1 ⊕ i K 2 ) ≲ ( m p r ) 2 m p 2 r − i = m i p i m i − 1 ≤ 1 m , \frac{\mathbf{P}^2}{\pi(\mathrm{K}_1\oplus_i\mathrm{K}_2)}\lesssim\frac{(mp^r)^2}{mp^{2r-i}}=\frac{m^ip^i}{m^{i-1}}\le\frac{1}{m}, π ( K 1 ⊕ i K 2 ) P 2 ≲ m p 2 r − i ( m p r ) 2 = m i − 1 m i p i ≤ m 1 , which gives us (43) in this range of parameters. If m p 2 ≥ 1 mp^2 \ge1 m p 2 ≥ 1 m p 3 ≤ 1 mp^3 \le1 m p 3 ≤ 1 (41) follows directly from Fact 4.2, as any subgraph probability is then comparable to a constant. Regarding (43) , by Lemma 4.21 and Fact 4.2 we get
π ( K 1 ⊕ i K 2 ) − P 2 = P 2 ( 1 π ( K i ) − 1 ) + O ( m p 3 e − 2 m p 2 ) = P 2 1 − π ( K i ) π ( K i ) + O ( ( 1 − p ) 2 ) ≍ 1 − p ≍ π ( K 1 ⊕ i K 2 ) ( 1 − p ) . \begin{aligned}
\pi(K_1 \oplus_i K_2)-\mathbf{P}^2 &= \mathbf{P}^2\left(\frac{1}{\pi(K_i)}-1\right)+O(mp^3e^{-2mp^2}) \\
&= \mathbf{P}^2\frac{1-\pi(K_i)}{\pi(K_i)}+O((1-\mathbf{p})^2) \\
&\asymp1-\mathbf{p} \\
&\asymp\pi(K_1 \oplus_i K_2)(1-\mathbf{p}).
\end{aligned} π ( K 1 ⊕ i K 2 ) − P 2 = P 2 ( π ( K i ) 1 − 1 ) + O ( m p 3 e − 2 m p 2 ) = P 2 π ( K i ) 1 − π ( K i ) + O (( 1 − p ) 2 ) ≍ 1 − p ≍ π ( K 1 ⊕ i K 2 ) ( 1 − p ) . We now turn our attention to the case when m p ≥ 1 mp \ge1 m p ≥ 1 m p 2 ≤ 1 mp^2 \le1 m p 2 ≤ 1
The upper bound follows directly from Lemma 4.9 and the fact that K 1 ⊕ i K 2 ≅ G 2 , I K_1 \oplus_i K_2 \cong G_{2,I} K 1 ⊕ i K 2 ≅ G 2 , I I = { 11 , 12 } I=\{11,12\} I = { 11 , 12 }
Now we address the lower bound. For m − 1 ≤ p ≤ m − r + 1 2 r m^{-1} \le p \le m^{-\frac{r+1}{2r}} m − 1 ≤ p ≤ m − 2 r r + 1 (9) , (10) , and Lemma 4.5 we have
n 2 r − i π ( K 1 ⊕ i K 2 ) ≥ n 2 r − i π ( K 1 ⊕ i K 2 , { V ( K 1 ⊕ i K ) } ) ≍ n 2 r − i m p 2 r − i = ( n r m p r ) 2 n i p i ≍ ( n r P ) 2 Ψ i . n^{2r-i}\pi(K_1\oplus_i K_2)\ge n^{2r-i}\pi(K_1\oplus_i K_2,\{V(K_1\oplus_i K)\})
\asymp n^{2r-i}mp^{2r-i}=\frac{(n^rmp^r)^2}{n^ip^i}\asymp\frac{(n^r\mathbf{P})^2}{\Psi_i}. n 2 r − i π ( K 1 ⊕ i K 2 ) ≥ n 2 r − i π ( K 1 ⊕ i K 2 , { V ( K 1 ⊕ i K )}) ≍ n 2 r − i m p 2 r − i = n i p i ( n r m p r ) 2 ≍ Ψ i ( n r P ) 2 . Similarly, for m − r + 1 2 r ≤ p ≤ m − 1 2 m^{-\frac{r+1}{2r}}\le p\le m^{-\frac{1}{2}} m − 2 r r + 1 ≤ p ≤ m − 2 1 (9) , (10) , and Lemma 4.5 we obtain
n 2 r − i π ( K 1 ⊕ i K 2 ) ≥ n 2 r − i π ( K 1 ⊕ i K 2 , E ( K 1 ⊕ i K 2 ) ) ≍ n 2 r − i p ( 2 2 ) − ( i 2 ) = ( n r P ) 2 Ψ i , n^{2r-i}\pi(K_1\oplus_i K_2)\ge n^{2r-i}\pi(K_1\oplus_i K_2,E(K_1\oplus_i K_2))\asymp n^{2r-i}p^{\binom{2}{2}-\binom{i}{2}}=\frac{(n^r\mathbf{P})^2}{\Psi_i}, n 2 r − i π ( K 1 ⊕ i K 2 ) ≥ n 2 r − i π ( K 1 ⊕ i K 2 , E ( K 1 ⊕ i K 2 )) ≍ n 2 r − i p ( 2 2 ) − ( 2 i ) = Ψ i ( n r P ) 2 , which concludes the proof of (42) .
Eventually, we examine the difference π ( K 1 ⊕ i K 2 ) − P 2 \pi(K_1\oplus_i K_2)-\mathbf{P}^2 π ( K 1 ⊕ i K 2 ) − P 2 m p ≥ 1 mp\ge1 m p ≥ 1 m p 2 ≤ 1 mp^2\le1 m p 2 ≤ 1 ∣ V ( K 1 ⊕ i K 2 ) ∣ = 2 r − i |V(K_1\oplus_i K_2)|=2r-i ∣ V ( K 1 ⊕ i K 2 ) ∣ = 2 r − i p ≤ m − ( 2 r − i ) + 1 2 ( 2 r − i ) + δ p\le m^{-\frac{(2r-i)+1}{2(2r-i)}+\delta} p ≤ m − 2 ( 2 r − i ) ( 2 r − i ) + 1 + δ 0 < δ < 1 / 8 r 0<\delta<1/8r 0 < δ < 1/8 r 2 ( 2 r − i ) < 4 r 2(2r-i)<4r 2 ( 2 r − i ) < 4 r
− ( 2 r − i ) + 1 2 ( 2 r − i ) + δ < − 1 2 − 1 2 ( 2 r − i ) + 1 8 r < − 1 2 − 1 8 r . -\frac{(2r-i)+1}{2(2r-i)}+\delta<-\frac{1}{2}-\frac{1}{2(2r-i)}+\frac{1}{8r}<-\frac{1}{2}-\frac{1}{8r}. − 2 ( 2 r − i ) ( 2 r − i ) + 1 + δ < − 2 1 − 2 ( 2 r − i ) 1 + 8 r 1 < − 2 1 − 8 r 1 . In particular, this implies
m p 2 ≤ m − 1 / ( 4 r ) . mp^2\le m^{-1/(4r)}. m p 2 ≤ m − 1/ ( 4 r ) . Then, for all 2 ≤ i ≤ r 2\le i\le r 2 ≤ i ≤ r
Ψ i ≲ n i ( m p i + ( m p 2 ) ( i 2 ) ) ≤ n i m p 2 ≤ n i m − 1 / ( 4 r ) . \Psi_i\lesssim n^i\left(mp^i+(mp^2)^{\binom{i}{2}}\right)\le n^imp^2\le n^im^{-1/(4r)}. Ψ i ≲ n i ( m p i + ( m p 2 ) ( 2 i ) ) ≤ n i m p 2 ≤ n i m − 1/ ( 4 r ) . Therefore, using the lower bound in (42) we get
P 2 ≲ n i m − 1 / ( 4 r ) Ψ i P 2 = m − 1 / ( 4 r ) n i − 2 r ( n r P ) 2 Ψ i ≲ m − 1 / ( 4 r ) π ( K 1 ⊕ i K 2 ) . \mathbf{P}^2\lesssim\frac{n^im^{-1/(4r)}}{\Psi_i}\mathbf{P}^2=m^{-1/(4r)}n^{i-2r}\frac{(n^r\mathbf{P})^2}{\Psi_i}\lesssim m^{-1/(4r)}\pi(K_1\oplus_i K_2). P 2 ≲ Ψ i n i m − 1/ ( 4 r ) P 2 = m − 1/ ( 4 r ) n i − 2 r Ψ i ( n r P ) 2 ≲ m − 1/ ( 4 r ) π ( K 1 ⊕ i K 2 ) . Thus, since p = o ( 1 ) p=o(1) p = o ( 1 )
π ( K 1 ⊕ i K 2 ) − P 2 = ( 1 + O ( m − 1 / ( 4 r ) ) ) π ( K 1 ⊕ i K 2 ) ≍ ( 1 − p ) π ( K 1 ⊕ i K 2 ) . \pi(K_1\mathbin{\oplus_i}K_2)-\mathbf{P}^2=(1+O(m^{-1/(4r)}))\pi(K_1\mathbin{\oplus_i}K_2)\asymp(1-p)\pi(K_1\mathbin{\oplus_i}K_2). π ( K 1 ⊕ i K 2 ) − P 2 = ( 1 + O ( m − 1/ ( 4 r ) )) π ( K 1 ⊕ i K 2 ) ≍ ( 1 − p ) π ( K 1 ⊕ i K 2 ) . This implies (43) for p ≤ m − ( 2 r − i ) + 1 2 ( 2 r − i ) + δ p\le m^{-\frac{(2r-i)+1}{2(2r-i)}+\delta} p ≤ m − 2 ( 2 r − i ) ( 2 r − i ) + 1 + δ 0 < δ < 1 / 8 r 0<\delta<1/8r 0 < δ < 1/8 r
Now we assume m − ( 2 r − i ) + 1 2 ( 2 r − i ) + δ ≤ p ≤ m − 1 / 2 m^{-\frac{(2r-i)+1}{2(2r-i)}+\delta}\le p\le m^{-1/2} m − 2 ( 2 r − i ) ( 2 r − i ) + 1 + δ ≤ p ≤ m − 1/2 p p p Lemma 4.11 (iii) with A = m δ A=m^\delta A = m δ 0 < δ < δ 2 r < δ r 0<\delta<\delta_{2r}<\delta_r 0 < δ < δ 2 r < δ r Lemma 4.11 (iii) gives us
P = π ( K ) = ( 1 + O ( m − δ + m − δ r ) ) p ( r 2 ) = ( 1 + O ( m − δ ) ) p ( r 2 ) . \mathbf{P}=\pi(K)=(1+O(m^{-\delta}+m^{-\delta_r}))p^{\binom{r}{2}}=(1+O(m^{-\delta}))p^{\binom{r}{2}}. P = π ( K ) = ( 1 + O ( m − δ + m − δ r )) p ( 2 r ) = ( 1 + O ( m − δ )) p ( 2 r ) . π ( K 1 ⊕ i K 2 ) = ( 1 + O ( m − δ + m − δ 2 r ) ) p ( r 2 ) − ( i 2 ) = ( 1 + O ( m − δ ) ) p ( r 2 ) − ( i 2 ) . \pi(K_1\mathbin{\oplus_i}K_2)=(1+O(m^{-\delta}+m^{-\delta_{2r}}))p^{\binom{r}{2}-\binom{i}{2}}=(1+O(m^{-\delta}))p^{\binom{r}{2}-\binom{i}{2}}. π ( K 1 ⊕ i K 2 ) = ( 1 + O ( m − δ + m − δ 2 r )) p ( 2 r ) − ( 2 i ) = ( 1 + O ( m − δ )) p ( 2 r ) − ( 2 i ) . Consequently, using the above equalities, we obtain for i ≥ 2 i\ge2 i ≥ 2
P 2 = ( 1 + O ( m − δ ) ) p ( r 2 ) = p ( i 2 ) ( 1 + O ( m − δ ) ) p ( r 2 ) − ( i 2 ) = p ( i 2 ) ( 1 + O ( m − δ ) ) π ( K 1 ⊕ i K 2 ) ≤ p ( 1 + O ( m − δ ) ) π ( K 1 ⊕ i K 2 ) . \begin{aligned}
\mathbf{P}^2=(1+O(m^{-\delta}))p^{\binom{r}{2}}=p^{\binom{i}{2}}(1+O(m^{-\delta}))p^{\binom{r}{2}-\binom{i}{2}} \\
&=p^{\binom{i}{2}}(1+O(m^{-\delta}))\pi(K_1\mathbin{\oplus_i}K_2) \\
&\le p(1+O(m^{-\delta}))\pi(K_1\mathbin{\oplus_i}K_2).
\end{aligned} P 2 = ( 1 + O ( m − δ )) p ( 2 r ) = p ( 2 i ) ( 1 + O ( m − δ )) p ( 2 r ) − ( 2 i ) = p ( 2 i ) ( 1 + O ( m − δ )) π ( K 1 ⊕ i K 2 ) ≤ p ( 1 + O ( m − δ )) π ( K 1 ⊕ i K 2 ) . Hence,
( 1 − p − O ( m − δ ) ) π ( K 1 ⊕ i K 2 ) ≤ π ( K 1 ⊕ i K 2 ) − P 2 ≤ π ( K 1 ⊕ i K 2 ) . (1-p-O(m^{-\delta}))\pi(K_1\mathbin{\oplus_i}K_2)\le\pi(K_1\mathbin{\oplus_i}K_2)-\mathbf{P}^2\le\pi(K_1\mathbin{\oplus_i}K_2). ( 1 − p − O ( m − δ )) π ( K 1 ⊕ i K 2 ) ≤ π ( K 1 ⊕ i K 2 ) − P 2 ≤ π ( K 1 ⊕ i K 2 ) . This, together with 1 − p 1-p 1 − p 0 0 0 m p 2 ≤ 1 mp^2\le1 m p 2 ≤ 1 (43) in this case. This completes the proof.
Lemma 5.2. There exists a constant c r > 0 c_r>0 c r > 0 r r r m p 3 ≤ c r mp^3\le c_r m p 3 ≤ c r m m m
( π ( K 1 ⊕ 1 K 2 ) − P 2 ) ≍ ( P p ) 2 m p 3 ( 1 − p ) 2 . \left(\pi(K_1\mathbin{\oplus_1}K_2)-\mathbf{P}^2\right)\asymp\left(\frac{\mathbf{P}}{p}\right)^2mp^3(1-p)^2. ( π ( K 1 ⊕ 1 K 2 ) − P 2 ) ≍ ( p P ) 2 m p 3 ( 1 − p ) 2 . Proof. Let us define two auxiliary graphs H ( 1 ) = K 1 ⊕ 1 K 2 ⊕ 0 N 1 H^{(1)}=K_1\mathbin{\oplus_1}K_2\mathbin{\oplus_0}N_1 H ( 1 ) = K 1 ⊕ 1 K 2 ⊕ 0 N 1 H ( 2 ) = K 1 ⊕ 0 K 2 H^{(2)}=K_1\mathbin{\oplus_0}K_2 H ( 2 ) = K 1 ⊕ 0 K 2 { K i } i ∈ Λ \{K_i\}_{i\in\Lambda} { K i } i ∈ Λ Λ = { 1 , 2 } \Lambda=\{1,2\} Λ = { 1 , 2 } C J ( H ( l ) ) C_J(H^{(l)}) C J ( H ( l ) ) J ⊆ Λ J\subseteq\Lambda J ⊆ Λ l = 1 , 2 l=1,2 l = 1 , 2 Definition 4.19 . Given l = 1 , 2 l=1,2 l = 1 , 2 C J ( H ( l ) ) C_J(H^{(l)}) C J ( H ( l ) ) J ⊆ Λ J\subseteq\Lambda J ⊆ Λ C ( H ( l ) ) = ⋃ J ⊆ Λ C J ( H ( l ) ) \mathcal{C}(H^{(l)})=\bigcup_{J\subseteq\Lambda}C_J(H^{(l)}) C ( H ( l ) ) = ⋃ J ⊆ Λ C J ( H ( l ) ) H { i } ( 1 ) ≅ K r ⊕ 0 N r ≅ H { i } ( 2 ) H_{\{i\}}^{(1)}\cong K_r\mathbin{\oplus_0}N_r\cong H_{\{i\}}^{(2)} H { i } ( 1 ) ≅ K r ⊕ 0 N r ≅ H { i } ( 2 ) i = 1 , 2 i=1,2 i = 1 , 2 H ∅ ( 1 ) ≅ N 2 r ≅ H ∅ ( 2 ) H_{\varnothing}^{(1)}\cong N_{2r}\cong H_{\varnothing}^{(2)} H ∅ ( 1 ) ≅ N 2 r ≅ H ∅ ( 2 ) (9) and Lemma 4.20
π ( K 1 ⊕ 1 K 2 ) − P 2 = π ( H ( 1 ) ) − π ( H ( 2 ) ) = ∑ J ⊆ Λ ( π ( H ( 1 ) , C J ( H ( 1 ) ) ) − π ( H ( 2 ) , C J ( H ( 2 ) ) ) ) = π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) − π ( H ( 2 ) , C Λ ( H ( 2 ) ) ) . \begin{aligned}
\pi(K_1\mathbin{\oplus_1}K_2)-\mathbf{P}^2=\pi(H^{(1)})-\pi(H^{(2)}) \\
&=\sum_{J\subseteq\Lambda}\left(\pi(H^{(1)},C_J(H^{(1)}))-\pi(H^{(2)},C_J(H^{(2)}))\right) \\
&=\pi(H^{(1)},C_\Lambda(H^{(1)}))-\pi(H^{(2)},C_\Lambda(H^{(2)})).
\end{aligned} π ( K 1 ⊕ 1 K 2 ) − P 2 = π ( H ( 1 ) ) − π ( H ( 2 ) ) = J ⊆ Λ ∑ ( π ( H ( 1 ) , C J ( H ( 1 ) )) − π ( H ( 2 ) , C J ( H ( 2 ) )) ) = π ( H ( 1 ) , C Λ ( H ( 1 ) )) − π ( H ( 2 ) , C Λ ( H ( 2 ) )) . Note that C Λ ( H ( l ) ) C_\Lambda(H^{(l)}) C Λ ( H ( l ) ) l = 1 , 2 l=1,2 l = 1 , 2 H ( l ) H^{(l)} H ( l ) l = 1 , 2 l=1,2 l = 1 , 2 { K i } i ∈ Λ \{K_i\}_{i\in\Lambda} { K i } i ∈ Λ Λ = { 1 , 2 } \Lambda=\{1,2\} Λ = { 1 , 2 } (39) , it now suffices to show
π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) − π ( H ( 2 ) , C Λ ( H ( 2 ) ) ≍ { P 2 m p , for m p 2 ≤ 1 ; m p 3 e − 2 m p 2 , for m p 2 ≥ 1 and m p 3 ≤ c r . (44) \pi(H^{(1)},C_\Lambda(H^{(1)}))-\pi(H^{(2)},C_\Lambda(H^{(2)})
\asymp
\begin{cases}
\dfrac{\mathbf{P}^2}{mp}, & \text{for }mp^2\le1;\\
mp^3e^{-2mp^2}, & \text{for }mp^2\ge1\text{ and }mp^3\le c_r.
\end{cases} \tag*{(44)} π ( H ( 1 ) , C Λ ( H ( 1 ) )) − π ( H ( 2 ) , C Λ ( H ( 2 ) ) ≍ ⎩ ⎨ ⎧ m p P 2 , m p 3 e − 2 m p 2 , for m p 2 ≤ 1 ; for m p 2 ≥ 1 and m p 3 ≤ c r . (44) First, we consider the case m p ≤ 1 mp \le1 m p ≤ 1 { V ( K 1 ⊕ 1 K 2 ) } ∈ C Λ ( H ( 1 ) ) \{V(\mathrm{K}_1 \oplus_1 \mathrm{K}_2)\} \in\mathcal{C}_{\Lambda}(H^{(1)}) { V ( K 1 ⊕ 1 K 2 )} ∈ C Λ ( H ( 1 ) ) K 1 ⊕ 1 K 2 \mathrm{K}_1 \oplus_1 \mathrm{K}_2 K 1 ⊕ 1 K 2
π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) ≍ π ( H ( 1 ) , { V ( K 1 ⊕ 1 K 2 ) } ) ≍ m p 2 r − 1 = ( m p r ) 2 m p ≍ P 2 m p . \pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(1)})) \asymp\pi(H^{(1)},\{V(\mathrm{K}_1 \oplus_1 \mathrm{K}_2)\}) \asymp mp^{2r-1} = \frac{(mp^r)^2}{mp} \asymp\frac{\mathbf{P}^2}{mp}. π ( H ( 1 ) , C Λ ( H ( 1 ) )) ≍ π ( H ( 1 ) , { V ( K 1 ⊕ 1 K 2 )}) ≍ m p 2 r − 1 = m p ( m p r ) 2 ≍ m p P 2 . On the other hand, by Lemma 4.22 with j = 1 j=1 j = 1 p ≤ m − 1 ⇔ m p ≤ 1 p \le m^{-1} \Leftrightarrow mp \le1 p ≤ m − 1 ⇔ m p ≤ 1
π ( H ( 2 ) , C Λ ( H ( 2 ) ) ) ≲ P 2 m ≲ m − 1 P 2 m p . (45) \pi(H^{(2)},\mathcal{C}_{\Lambda}(H^{(2)})) \lesssim\frac{\mathbf{P}^2}{m} \lesssim m^{-1}\frac{\mathbf{P}^2}{mp}. \tag*{(45)} π ( H ( 2 ) , C Λ ( H ( 2 ) )) ≲ m P 2 ≲ m − 1 m p P 2 . (45) Therefore
π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) − π ( H ( 2 ) , C Λ ( H ( 2 ) ) ) ≍ ( 1 + O ( m − 1 ) ) P 2 m p ≲ P 2 m p . \pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(1)}))-\pi(H^{(2)},\mathcal{C}_{\Lambda}(H^{(2)})) \asymp(1+O(m^{-1}))\frac{\mathbf{P}^2}{mp} \lesssim\frac{\mathbf{P}^2}{mp}. π ( H ( 1 ) , C Λ ( H ( 1 ) )) − π ( H ( 2 ) , C Λ ( H ( 2 ) )) ≍ ( 1 + O ( m − 1 )) m p P 2 ≲ m p P 2 . which implies (44) for m p ≤ 1 mp \le1 m p ≤ 1
In the case m − 1 ≤ p ≤ m − 1 / 2 m^{-1} \le p \le m^{-1/2} m − 1 ≤ p ≤ m − 1/2 H ( 1 ) H^{(1)} H ( 1 ) Lemma 4.25 . We recall that we consider partition { K i } i ∈ Λ \{K_i\}_{i\in\Lambda} { K i } i ∈ Λ Λ = { 1 , 2 } \Lambda=\{1,2\} Λ = { 1 , 2 } F = F ( H ( 1 ) , { K i } i ∈ Λ ) F=F(H^{(1)},\{K_i\}_{i\in\Lambda}) F = F ( H ( 1 ) , { K i } i ∈ Λ ) Lemma 4.25
π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) ≲ P 2 m p . \pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(1)})) \lesssim\frac{\mathbf{P}^2}{mp}. π ( H ( 1 ) , C Λ ( H ( 1 ) )) ≲ m p P 2 . To obtain the lower bound, we consider two indecomposable clique covers. The first one is simply { V ( K 1 ⊕ 1 K 2 ) } \{V(\mathrm{K}_1\oplus_1\mathrm{K}_2)\} { V ( K 1 ⊕ 1 K 2 )} v ∈ V ( K 1 ) ∩ V ( K 2 ) v\in V(\mathrm{K}_1)\cap V(\mathrm{K}_2) v ∈ V ( K 1 ) ∩ V ( K 2 ) v i ∈ V ( K i ) ∖ { v } v_i\in V(K_i)\setminus\{v\} v i ∈ V ( K i ) ∖ { v } C ∩ \mathcal{C}_{\cap} C ∩ E ( H ( 1 ) ) E(H^{(1)}) E ( H ( 1 ) ) { v , v 1 } \{v,v_1\} { v , v 1 } { v , v 2 } \{v,v_2\} { v , v 2 } { v , v 1 , v 2 } \{v,v_1,v_2\} { v , v 1 , v 2 }
C ∩ = ( E ( H ( 1 ) ) ∪ { v , v 1 , v 2 } ) ∖ { { v , v 1 } , { v , v 2 } } . (46) \mathcal{C}_{\cap}=\left(E(H^{(1)})\cup\{v,v_1,v_2\}\right)\setminus\{\{v,v_1\},\{v,v_2\}\}. \tag*{(46)} C ∩ = ( E ( H ( 1 ) ) ∪ { v , v 1 , v 2 } ) ∖ {{ v , v 1 } , { v , v 2 }} . (46) The clique cover C ∩ \mathcal{C}_{\cap} C ∩ (10) and Lemma 4.5 we get for p ≤ m − r + 1 2 r p\le m^{-\frac{r+1}{2r}} p ≤ m − 2 r r + 1
π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) ≥ π ( H ( 1 ) , { V ( H ( 1 ) ) } ) ≍ n 2 r − 1 m p 2 r − 1 ≲ P 2 m p , \pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(1)}))\ge\pi(H^{(1)},\{V(H^{(1)})\})\asymp n^{2r-1}mp^{2r-1}\lesssim\frac{\mathbf{P}^2}{mp}, π ( H ( 1 ) , C Λ ( H ( 1 ) )) ≥ π ( H ( 1 ) , { V ( H ( 1 ) )}) ≍ n 2 r − 1 m p 2 r − 1 ≲ m p P 2 , whereas for m − r + 1 2 r ≤ p ≤ m − 1 / 2 m^{-\frac{r+1}{2r}}\le p\le m^{-1/2} m − 2 r r + 1 ≤ p ≤ m − 1/2
π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) ≥ π ( H ( 1 ) , C ∩ ) ≍ m p 3 ( m p 2 ) 2 ( r ) 2 − 2 ≲ P 2 m p . \pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(1)}))\ge\pi(H^{(1)},\mathcal{C}_{\cap})\asymp mp^3(mp^2)^{\frac{2(r)}{2}-2}\lesssim\frac{\mathbf{P}^2}{mp}. π ( H ( 1 ) , C Λ ( H ( 1 ) )) ≥ π ( H ( 1 ) , C ∩ ) ≍ m p 3 ( m p 2 ) 2 2 ( r ) − 2 ≲ m p P 2 . Eventually, analogously as in (45) , Lemma 4.22 yields π ( H ( 2 ) , C Λ ( H ( 2 ) ) ) = O ( P 2 / m ) = O ( m − 1 / 2 P 2 / m p ) \pi(H^{(2)},\mathcal{C}_{\Lambda}(H^{(2)}))=O(\mathbf{P}^2/m)=O(m^{-1/2}\mathbf{P}^2/mp) π ( H ( 2 ) , C Λ ( H ( 2 ) )) = O ( P 2 / m ) = O ( m − 1/2 P 2 / m p )
π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) − π ( H ( 2 ) , C Λ ( H ( 2 ) ) ) ≍ ( 1 + O ( m − 1 / 2 ) ) P 2 m p , \pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(1)}))-\pi(H^{(2)},\mathcal{C}_{\Lambda}(H^{(2)}))\asymp(1+O(m^{-1/2}))\frac{\mathbf{P}^2}{mp}, π ( H ( 1 ) , C Λ ( H ( 1 ) )) − π ( H ( 2 ) , C Λ ( H ( 2 ) )) ≍ ( 1 + O ( m − 1/2 )) m p P 2 , as required.
Now, let us consider the case m p 2 ≥ 1 mp^{2} \ge1 m p 2 ≥ 1 m p 3 ≤ 1 mp^{3} \le1 m p 3 ≤ 1 (8) we obtain
π ( H ( 1 ) , C Λ ( H ( 2 ) ) ) ≲ m p 3 e − 2 m p 2 , π ( H ( 2 ) , C Λ ( H ( 2 ) ) ) ≲ ( m p 3 ) 2 e − 2 m p 2 . \begin{aligned}
\pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(2)})) &\lesssim mp^{3}e^{-2mp^{2}},\\
\pi(H^{(2)},\mathcal{C}_{\Lambda}(H^{(2)})) &\lesssim(mp^{3})^{2}e^{-2mp^{2}}.
\end{aligned} π ( H ( 1 ) , C Λ ( H ( 2 ) )) π ( H ( 2 ) , C Λ ( H ( 2 ) )) ≲ m p 3 e − 2 m p 2 , ≲ ( m p 3 ) 2 e − 2 m p 2 . On the other hand, for C ∩ \mathcal{C}_{\cap} C ∩ (46) , by (12) we have
π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) ≥ π ( H ( 1 ) , C ∩ ) ≳ m p 3 e − 2 m p 2 . \pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(1)})) \ge\pi(H^{(1)},\mathcal{C}_{\cap}) \gtrsim mp^{3}e^{-2mp^{2}}. π ( H ( 1 ) , C Λ ( H ( 1 ) )) ≥ π ( H ( 1 ) , C ∩ ) ≳ m p 3 e − 2 m p 2 . This gives us
m p 3 e − 2 m p 2 ≳ π ( H ( 1 ) , C Λ ( H ( 1 ) ) ) − π ( H ( 2 ) , C Λ ( H ( 2 ) ) ) ≳ m p 3 e − 2 m p 2 ( 1 + O ( m p 3 ) ) , mp^{3}e^{-2mp^{2}} \gtrsim\pi(H^{(1)},\mathcal{C}_{\Lambda}(H^{(1)}))-\pi(H^{(2)},\mathcal{C}_{\Lambda}(H^{(2)})) \gtrsim mp^{3}e^{-2mp^{2}}(1+O(mp^{3})), m p 3 e − 2 m p 2 ≳ π ( H ( 1 ) , C Λ ( H ( 1 ) )) − π ( H ( 2 ) , C Λ ( H ( 2 ) )) ≳ m p 3 e − 2 m p 2 ( 1 + O ( m p 3 )) , which implies (44) for sufficiently small c r c_r c r c r c_r c r r r r 2 r 2r 2 r
Proof of Theorem 2.2
The main strength of Theorem 2.2 lies in the bound (4) . In order to fully concentrate on its proof, we first deal with the remaining parts of the assertion of the theorem.
Lemma 6.1. Assume m p 3 ≤ c r mp^{3} \le c_r m p 3 ≤ c r c r c_r c r
X ~ n ( r ) → D N ⇒ ( 1 − p ) Ψ → ∞ . \widetilde{X}_n^{(r)} \xrightarrow{\mathrm{D}} \mathcal{N} \Rightarrow(1-p)\Psi\to\infty. X n ( r ) D N ⇒ ( 1 − p ) Ψ → ∞. Proof. We adapt the argument from [16 proof of Theorem 3.5] . By Theorem 2.1, for m p 3 ≤ c r mp^{3} \le c_r m p 3 ≤ c r c = c ( r ) c=c(r) c = c ( r )
X ~ n ( r ) = X n ( r ) − E [ X n ( r ) ] Var [ X n ( r ) ] ≥ − E [ X n ( r ) ] Var [ X n ( r ) ] ≥ − c n r P ( 1 − p ) ( n r P ) 2 / Ψ = − c Ψ 1 − p . \widetilde{X}_n^{(r)}=\frac{X_n^{(r)}-\mathbb{E}[X_n^{(r)}]}{\sqrt{\operatorname{Var}[X_n^{(r)}]}} \ge-\frac{\mathbb{E}[X_n^{(r)}]}{\sqrt{\operatorname{Var}[X_n^{(r)}]}} \ge-c\frac{n^r\mathbf{P}}{\sqrt{(1-p)(n^r\mathbf{P})^2/\Psi}}=-c\sqrt{\frac{\Psi}{1-p}}. X n ( r ) = Var [ X n ( r ) ] X n ( r ) − E [ X n ( r ) ] ≥ − Var [ X n ( r ) ] E [ X n ( r ) ] ≥ − c ( 1 − p ) ( n r P ) 2 /Ψ n r P = − c 1 − p Ψ . Since the normal distribution is supported on the whole real line, the last expression has to tend to minus infinity, which follows Ψ / ( 1 − p ) → ∞ \Psi/(1-p)\to\infty Ψ/ ( 1 − p ) → ∞ p ≤ 1 / 2 p\le1/2 p ≤ 1/2
Consider now p ≥ 1 / 2 p\ge1/2 p ≥ 1/2 V n ( r ) = ( n r ) − X n ( r ) V_n^{(r)}=\binom{n}{r}-X_n^{(r)} V n ( r ) = ( r n ) − X n ( r ) K r K_r K r G ( n , m , p ) G(n,m,p) G ( n , m , p ) Var [ V n ( r ) ] = Var [ X n ( r ) ] \operatorname{Var}[V_n^{(r)}]=\operatorname{Var}[X_n^{(r)}] Var [ V n ( r ) ] = Var [ X n ( r ) ] V ~ n ( r ) = − X ~ n ( r ) → D N \widetilde{V}_n^{(r)}=-\widetilde{X}_n^{(r)}\xrightarrow{\mathrm{D}}\mathcal{N} V n ( r ) = − X n ( r ) D N P ≍ 1 \mathbf{P}\asymp1 P ≍ 1 ( 1 − P ) ≍ ( 1 − p ) (1-\mathbf{P})\asymp(1-p) ( 1 − P ) ≍ ( 1 − p ) c ′ = c ′ ( r ) c'=c'(r) c ′ = c ′ ( r ) n n n
V ~ n ( r ) = V n ( r ) − E [ V n ( r ) ] Var [ V n ( r ) ] ≥ − E [ V n ( r ) ] Var [ V n ( r ) ] ≥ − c ( n r ) ( 1 − P ) ( 1 − p ) ( n r P ) 2 / Ψ ≥ − c ′ ( 1 − p ) Ψ , \widetilde{V}_n^{(r)}=\frac{V_n^{(r)}-\mathbb{E}[V_n^{(r)}]}{\sqrt{\operatorname{Var}[V_n^{(r)}]}} \ge-\frac{\mathbb{E}[V_n^{(r)}]}{\sqrt{\operatorname{Var}[V_n^{(r)}]}} \ge-c\frac{\binom{n}{r}(1-\mathbf{P})}{\sqrt{(1-\mathbf{p})(n^r\mathbf{P})^2/\Psi}} \ge-c'\sqrt{(1-p)\Psi}, V n ( r ) = Var [ V n ( r ) ] V n ( r ) − E [ V n ( r ) ] ≥ − Var [ V n ( r ) ] E [ V n ( r ) ] ≥ − c ( 1 − p ) ( n r P ) 2 /Ψ ( r n ) ( 1 − P ) ≥ − c ′ ( 1 − p ) Ψ , which follows ( 1 − p ) Ψ → ∞ (1-p)\Psi\to\infty ( 1 − p ) Ψ → ∞
Lemma 6.2. We have
[ ( 1 ( 1 − p ) Ψ ) + 1 n + 1 n m p ] 1 / 2 ∧ 1 ≲ ( 1 ( 1 − p ) Ψ ) 1 / 4 . \left[\left(\frac{1}{(1-p)\Psi}\right)+\frac{1}{n}+\frac{1}{nmp}\right]^{1/2}\wedge1\lesssim\left(\frac{1}{(1-p)\Psi}\right)^{1/4}. [ ( ( 1 − p ) Ψ 1 ) + n 1 + nm p 1 ] 1/2 ∧ 1 ≲ ( ( 1 − p ) Ψ 1 ) 1/4 . Proof. Since x 1 / 2 ∧ 1 ≤ x 1 / 4 x^{1/2}\wedge1\le x^{1/4} x 1/2 ∧ 1 ≤ x 1/4 x ≥ 0 x\ge0 x ≥ 0
[ ( 1 ( 1 − p ) Ψ ) + 1 n + 1 n m p ] 1 / 2 ∧ 1 ≤ [ ( 1 ( 1 − p ) Ψ ) 1 / 2 ∧ 1 ] + ( 1 n ) 1 / 2 + ( 1 n m p ) 1 / 2 ≤ ( 1 ( 1 − p ) Ψ ) 1 / 4 + ( 1 n 2 p ) 1 / 4 + ( 1 m ( n 2 ( m p 2 ) ) ) 1 / 4 \begin{aligned}
\left[\left(\frac{1}{(1-p)\Psi}\right)+\frac{1}{n}+\frac{1}{nmp}\right]^{1/2}\wedge1\le\left[\left(\frac{1}{(1-p)\Psi}\right)^{1/2}\wedge1\right]+\left(\frac{1}{n}\right)^{1/2}+\left(\frac{1}{nmp}\right)^{1/2} \\
&\le\left(\frac{1}{(1-p)\Psi}\right)^{1/4}+\left(\frac{1}{n^2p}\right)^{1/4}+\left(\frac{1}{m\left(n^2(mp^2)\right)}\right)^{1/4}
\end{aligned} [ ( ( 1 − p ) Ψ 1 ) + n 1 + nm p 1 ] 1/2 ∧ 1 ≤ [ ( ( 1 − p ) Ψ 1 ) 1/2 ∧ 1 ] + ( n 1 ) 1/2 + ( nm p 1 ) 1/2 ≤ ( ( 1 − p ) Ψ 1 ) 1/4 + ( n 2 p 1 ) 1/4 + ( m ( n 2 ( m p 2 ) ) 1 ) 1/4 ≲ ( 1 ( 1 − p ) Ψ ) 1 / 4 , \lesssim\left(\frac{1}{(1-p)\Psi}\right)^{1/4}, ≲ ( ( 1 − p ) Ψ 1 ) 1/4 , where we also used m p 2 ≳ p mp^2\gtrsim p m p 2 ≳ p (1) , and Ψ ≤ n 2 p \Psi\le n^2p Ψ ≤ n 2 p
In order to prove Theorem 2.2 , it now suffices to establish the bound (4) . Indeed, since convergence of the distance between X n ( r ) X_n^{(r)} X n ( r ) N \mathcal{N} N X n ( r ) → D N X_n^{(r)}\xrightarrow{D}\mathcal{N} X n ( r ) D N (4) , together with Lemma 6.1, yields the asymptotic normality characterization. Furthermore, since d K / W ( X , Y ) ≤ 2 d_{K/W}(X,Y)\le2 d K / W ( X , Y ) ≤ 2 X X X Y Y Y (4) , combined with Lemma 6.2, gives the estimate (5) .
Thus, in view of Lemma 3.2, Theorem 2.2 follows from Lemma 6.3 stated below. We devote the remainder of this section to its proof.
Lemma 6.3. Let G t = G t , i , j G_t=G_{t,i,j} G t = G t , i , j t = 1 t=1 t = 1 0 ≤ i < j ≤ r 0\le i<j\le r 0 ≤ i < j ≤ r t = 2 , 3 t=2,3 t = 2 , 3 1 ≤ i < j ≤ r 1\le i<j\le r 1 ≤ i < j ≤ r m p 3 ≤ c r mp^3\le c_r m p 3 ≤ c r
n ∣ V ( G t ) ∣ ∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) ≲ ( 1 ( 1 − p ) Ψ + 1 n + 1 n m p ) ( Var N ) 2 . (47) n^{|V(G_t)|}\sum_{I\subseteq\Gamma}(-1)^{|I|}\mathbf{P}^{|\Gamma|-|I|}\pi(G_{t,I})\lesssim\left(\frac{1}{(1-p)\Psi}+\frac{1}{n}+\frac{1}{nmp}\right)(\operatorname{Var}N)^2. \tag*{(47)} n ∣ V ( G t ) ∣ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G t , I ) ≲ ( ( 1 − p ) Ψ 1 + n 1 + nm p 1 ) ( Var N ) 2 . (47) We remark that from the method of the proof of Lemma 3.2 (see [22 ] ) it follows that the sum above is nonnegative. We split the proof into several cases. Each case is considered in a separate subsection. In some cases a very delicate estimation will be required. However, it will be often sufficient to use the following reasoning. For and any I ⊆ Γ I\subseteq\Gamma I ⊆ Γ
P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) ≤ π ( G t ) , \mathbf{P}^{|\Gamma|-|I|}\pi(G_{t,I})\le\pi(G_t), P ∣Γ∣ − ∣ I ∣ π ( G t , I ) ≤ π ( G t ) , which follows
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) ≤ ∑ I ⊆ Γ P ∣ Γ ∣ − ∣ I ∣ π ( G t ) ≤ 2 ∣ Γ ∣ π ( G t ) ≲ π ( G t ) . (48) \sum_{I\subseteq\Gamma}(-1)^{|I|}\mathbf{P}^{|\Gamma|-|I|}\pi(G_{t,I})\le\sum_{I\subseteq\Gamma}\mathbf{P}^{|\Gamma|-|I|}\pi(G_t)\le2^{|\Gamma|}\pi(G_t)\lesssim\pi(G_t). \tag*{(48)} I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G t , I ) ≤ I ⊆ Γ ∑ P ∣Γ∣ − ∣ I ∣ π ( G t ) ≤ 2 ∣Γ∣ π ( G t ) ≲ π ( G t ) . (48) Let us also note that (see Figure 2 )
∣ V ( G 1 ) ∣ = 4 r − 3 j + i , ∣ V ( G 2 ) ∣ = 4 r − 2 j , ∣ V ( G 3 ) ∣ = 4 r − j − i . (49) \begin{aligned}
|V(G_1)|&=4r-3j+i,\\
|V(G_2)|&=4r-2j,\\
|V(G_3)|&=4r-j-i. \tag*{(49)}
\end{aligned} ∣ V ( G 1 ) ∣ ∣ V ( G 2 ) ∣ ∣ V ( G 3 ) ∣ = 4 r − 3 j + i , = 4 r − 2 j , = 4 r − j − i . (49) The outline of the proof of Lemma 6.3 is as follows. First, in Section 6.1, we fully address the case m p ≤ 1 mp \le1 m p ≤ 1 m p ≥ 1 mp \ge1 m p ≥ 1
any two graphs K a b \mathbf{K}_{ab} K ab a b ∈ Γ ab \in\Gamma ab ∈ Γ § 6.4 ),
the graphs G 3 , 1 , j G_{3,1,j} G 3 , 1 , j 3 ≤ j ≤ r 3 \le j \le r 3 ≤ j ≤ r § 6.5 ).
The remaining cases are considered separately: for m p 2 ≤ 1 mp^2 \le1 m p 2 ≤ 1 m p 2 ≥ 1 mp^2 \ge1 m p 2 ≥ 1 § 6.3 .
Case m p ≤ 1 mp \le1 m p ≤ 1
For t = 1 t=1 t = 1 0 ≤ i < j ≤ r − 1 0 \le i < j \le r-1 0 ≤ i < j ≤ r − 1 Theorem 2.1 we have
n 4 r − 3 j + i π ( G 1 ) ≲ n 4 r − 3 j + i m p 4 r − 3 j + i ≲ ( n r m p r ) 4 ( n j m p j ) 2 n j − i m p j − i ≲ ( n r P ) 4 Ψ j 2 Ψ j − i ≲ ( Var N ) 2 Ψ . n^{4r-3j+i}\pi(G_1) \lesssim n^{4r-3j+i}mp^{4r-3j+i} \lesssim\frac{(n^rmp^r)^4}{(n^jmp^j)^2n^{j-i}mp^{j-i}} \lesssim\frac{(n^r\mathbf{P})^4}{\Psi_j^2\Psi_{j-i}} \lesssim\frac{(\operatorname{Var} N)^2}{\Psi}. n 4 r − 3 j + i π ( G 1 ) ≲ n 4 r − 3 j + i m p 4 r − 3 j + i ≲ ( n j m p j ) 2 n j − i m p j − i ( n r m p r ) 4 ≲ Ψ j 2 Ψ j − i ( n r P ) 4 ≲ Ψ ( Var N ) 2 . Similarly, for t = 2 , 3 t=2,3 t = 2 , 3
n 4 r − 2 j π ( G 2 ) ≲ n 4 r − 2 j m p 4 r − 2 j ≲ ( n r P ) 4 Ψ j Ψ j − i Ψ i ≲ ( Var N ) 2 Ψ , n^{4r-2j}\pi(G_2) \lesssim n^{4r-2j}mp^{4r-2j} \lesssim\frac{(n^r\mathbf{P})^4}{\Psi_j\Psi_{j-i}\Psi_i} \lesssim\frac{(\operatorname{Var} N)^2}{\Psi}, n 4 r − 2 j π ( G 2 ) ≲ n 4 r − 2 j m p 4 r − 2 j ≲ Ψ j Ψ j − i Ψ i ( n r P ) 4 ≲ Ψ ( Var N ) 2 , and
n 4 r − j − i π ( G 3 ) ≲ n 4 r − j − i m p 4 r − j − i ≲ ( n r P ) 4 Ψ j − i Ψ i 2 ≲ ( Var N ) 2 Ψ . n^{4r-j-i}\pi(G_3) \lesssim n^{4r-j-i}mp^{4r-j-i} \lesssim\frac{(n^r\mathbf{P})^4}{\Psi_{j-i}\Psi_i^2} \lesssim\frac{(\operatorname{Var} N)^2}{\Psi}. n 4 r − j − i π ( G 3 ) ≲ n 4 r − j − i m p 4 r − j − i ≲ Ψ j − i Ψ i 2 ( n r P ) 4 ≲ Ψ ( Var N ) 2 . In view of (48) , this proves Lemma 6.3 for m p ≤ 1 mp \le1 m p ≤ 1
No all-single intersections and m p ≥ 1 mp \ge1 m p ≥ 1 m p 2 ≤ 1 mp^2 \le1 m p 2 ≤ 1
No single intersections
Here we assume that if two cliques K a b \mathbf{K}_{ab} K ab a b ∈ Γ ab \in\Gamma ab ∈ Γ G t , i , j G_{t,i,j} G t , i , j
t = 1 , t = 2 , 3 , i ≥ 0 , i ≥ 2 , i + 2 ≤ j ≤ r , i + 2 ≤ j ≤ r . \begin{aligned}
t&=1, & t&=2,3,\\
i&\ge0, & i&\ge2,\\
i+2&\le j\le r, & i+2&\le j\le r.
\end{aligned} t i i + 2 = 1 , ≥ 0 , ≤ j ≤ r , t i i + 2 = 2 , 3 , ≥ 2 , ≤ j ≤ r . We will show that
n ∣ V ( G t ) ∣ π ( G t , C ) ≲ ( n r P ) 4 Ψ 3 . (50) n^{|V(G_t)|}\pi(G_t,\mathcal{C}) \lesssim\frac{(n^r\mathbf{P})^4}{\Psi^3}. \tag*{(50)} n ∣ V ( G t ) ∣ π ( G t , C ) ≲ Ψ 3 ( n r P ) 4 . (50) In view of (9) , (48) , and Theorem 2.1 , this proves Lemma 6.3 in this case.
We will use Lemma 4.9 three times. First, by Lemma 4.9 , there exists a connected G t , I G_{t,I} G t , I ∣ I ∣ = 3 |I|=3 ∣ I ∣ = 3
n ∣ V ( G t ) ∣ π ( G t , C ) = n ∣ V ( G t , Γ ) ∣ π ( G t , Γ , C ) ≲ n r P n ∣ V ( G t , I ) ∣ π ( G t , I ) Ψ . n^{|V(G_t)|}\pi(G_t,\mathcal{C})=n^{|V(G_{t,\Gamma})|}\pi(G_{t,\Gamma},\mathcal{C})\lesssim\frac{n^r\mathbf{P}n^{|V(G_{t,I})|}\pi(G_{t,I})}{\Psi}. n ∣ V ( G t ) ∣ π ( G t , C ) = n ∣ V ( G t , Γ ) ∣ π ( G t , Γ , C ) ≲ Ψ n r P n ∣ V ( G t , I ) ∣ π ( G t , I ) . Given t , i , j t,i,j t , i , j G t , I G_{t,\mathrm{I}} G t , I ∣ I ∣ = 3 |\mathrm{I}|=3 ∣ I ∣ = 3 C ∈ C ( G t ) \mathcal{C}\in\mathcal{C}(G_t) C ∈ C ( G t ) (9) we obtain
n ∣ V ( G t ) ∣ π ( G t ) ≲ n r P n ∣ V ( G t , { 11 , 12 , 21 } ) ∣ π ( G t , { 11 , 12 , 21 } ) Ψ . (51) n^{|V(G_t)|}\pi(G_t)\lesssim\frac{n^r\mathbf{P}n^{|V(G_{t,\{11,12,21\}})|}\pi(G_{t,\{11,12,21\}})}{\Psi}. \tag*{(51)} n ∣ V ( G t ) ∣ π ( G t ) ≲ Ψ n r P n ∣ V ( G t , { 11 , 12 , 21 } ) ∣ π ( G t , { 11 , 12 , 21 } ) . (51) Similarly, by Lemma 4.9 , for any C ∈ C ( G t , { 11 , 12 , 21 } ) \mathcal{C}\in\mathcal{C}(G_{t,\{11,12,21\}}) C ∈ C ( G t , { 11 , 12 , 21 } ) I ⊆ { 11 , 12 , 21 } \mathrm{I}\subseteq\{11,12,21\} I ⊆ { 11 , 12 , 21 } ∣ I ∣ = 2 |\mathrm{I}|=2 ∣ I ∣ = 2 G t , I G_{t,\mathrm{I}} G t , I
n ∣ V ( G t , { 11 , 12 , 21 } ) ∣ π ( G t , { 11 , 12 , 21 } , C ) ≲ n r P n ∣ V ( G t , I ) ∣ π ( G t , I ) Ψ . (52) n^{|V(G_{t,\{11,12,21\}})|}\pi(G_{t,\{11,12,21\}},\mathcal{C})\lesssim\frac{n^r\mathbf{P}n^{|V(G_{t,\mathrm{I}})|}\pi(G_{t,\mathrm{I}})}{\Psi}. \tag*{(52)} n ∣ V ( G t , { 11 , 12 , 21 } ) ∣ π ( G t , { 11 , 12 , 21 } , C ) ≲ Ψ n r P n ∣ V ( G t , I ) ∣ π ( G t , I ) . (52) Moreover, for any I ⊆ Γ \mathrm{I}\subseteq\Gamma I ⊆ Γ ∣ I ∣ = 2 |\mathrm{I}|=2 ∣ I ∣ = 2 G t , I G_{t,\mathrm{I}} G t , I Lemma 4.9 and (9) , we have
n ∣ V ( G t , I ) ∣ π ( G t , I ) = ∑ C ∈ C ( G t , I ) n ∣ V ( G t , I ) ∣ π ( G t , I , C ) ≲ ( n r P ) 2 Ψ . n^{|V(G_{t,\mathrm{I}})|}\pi(G_{t,\mathrm{I}})=\sum_{\mathcal{C}\in\mathcal{C}(G_{t,\mathrm{I}})}n^{|V(G_{t,\mathrm{I}})|}\pi(G_{t,\mathrm{I}},\mathcal{C})\lesssim\frac{(n^r\mathbf{P})^2}{\Psi}. n ∣ V ( G t , I ) ∣ π ( G t , I ) = C ∈ C ( G t , I ) ∑ n ∣ V ( G t , I ) ∣ π ( G t , I , C ) ≲ Ψ ( n r P ) 2 . Combining this with (52) , we obtain for all C ∈ C ( G t , { 11 , 12 , 21 } ) \mathcal{C}\in\mathcal{C}(G_{t,\{11,12,21\}}) C ∈ C ( G t , { 11 , 12 , 21 } )
n ∣ V ( G t , { 11 , 12 , 21 } ) ∣ π ( G t , { 11 , 12 , 21 } , C ) ≲ ( n r P ) 3 Ψ 2 , n^{|V(G_{t,\{11,12,21\}})|}\pi(G_{t,\{11,12,21\}},\mathcal{C})\lesssim\frac{(n^r\mathbf{P})^3}{\Psi^2}, n ∣ V ( G t , { 11 , 12 , 21 } ) ∣ π ( G t , { 11 , 12 , 21 } , C ) ≲ Ψ 2 ( n r P ) 3 , and consequently, by (9) , we have
n ∣ V ( G t , { 11 , 12 , 21 } ) ∣ π ( G t , { 11 , 12 , 21 } ) ≲ ( n r P ) 3 Ψ 2 . (53) n^{|V(G_{t,\{11,12,21\}})|}\pi(G_{t,\{11,12,21\}})\lesssim\frac{(n^r\mathbf{P})^3}{\Psi^2}. \tag*{(53)} n ∣ V ( G t , { 11 , 12 , 21 } ) ∣ π ( G t , { 11 , 12 , 21 } ) ≲ Ψ 2 ( n r P ) 3 . (53) Finally, (50) follows by (51) and (53) .
Both: single and non single intersections
Here we consider the cases when
t = 1 , i ≥ 1 , j = i + 1 ≤ r − 1 , t = 2 , 3 , i ≥ 2 , j = i + 1 ≤ r , t = 2 , i = 1 , i + 2 ≤ j ≤ r . \begin{aligned}
t&=1, & i&\ge1, & j&=i+1\le r-1,\\
t&=2,3, & i&\ge2, & j&=i+1\le r,\\
t&=2, & i&=1, & i+2\le j\le r.
\end{aligned} t t t = 1 , = 2 , 3 , = 2 , i i i ≥ 1 , ≥ 2 , = 1 , j j i + 2 ≤ j ≤ r . = i + 1 ≤ r − 1 , = i + 1 ≤ r , Since G 2 , j − 1 , j ≅ G 2 , 1 , j G_{2,j-1,j}\cong G_{2,1,j} G 2 , j − 1 , j ≅ G 2 , 1 , j j = i + 1 j=i+1 j = i + 1
As previously, we use (48) , so it suffices to estimate the probabilities π ( G t ) \pi(G_t) π ( G t ) m p 2 ≤ 1 mp^2\le1 m p 2 ≤ 1 m p ≥ 1 mp\ge1 m p ≥ 1 (9) , first two inequalities of Lemma 4.15 , the variance representation (38) and Lemma 5.1 , we get
n 4 r − 2 ( i + 1 ) − 1 π ( G 1 , i , i + 1 ) ≲ n 2 r − ( i + 1 ) π ( K 11 ⊕ i + 1 K 12 ) n 2 r − ( i + 1 ) π ( K 21 ⊕ i + 1 K 22 ) n ≲ ( Var N ) 2 n , i ≥ 1 , n^{4r-2(i+1)-1}\pi(G_{1,i,i+1})\lesssim\frac{n^{2r-(i+1)}\pi(K_{11}\mathbin{\oplus_{i+1}}K_{12})n^{2r-(i+1)}\pi(K_{21}\mathbin{\oplus_{i+1}}K_{22})}{n}\lesssim\frac{(\operatorname{Var}N)^2}{n},\qquad i\ge1, n 4 r − 2 ( i + 1 ) − 1 π ( G 1 , i , i + 1 ) ≲ n n 2 r − ( i + 1 ) π ( K 11 ⊕ i + 1 K 12 ) n 2 r − ( i + 1 ) π ( K 21 ⊕ i + 1 K 22 ) ≲ n ( Var N ) 2 , i ≥ 1 , n 4 r − 2 − 2 i π ( G 2 , i , i + 1 ) ≲ n 2 r − i π ( K 11 ⊕ i K 12 ) n 2 r − i π ( K 21 ⊕ i K 22 ) n 2 m p 2 ≲ ( Var N ) 2 Ψ , i ≥ 2 , n^{4r-2-2i}\pi(G_{2,i,i+1})\lesssim\frac{n^{2r-i}\pi(K_{11}\mathbin{\oplus_i}K_{12})n^{2r-i}\pi(K_{21}\mathbin{\oplus_i}K_{22})}{n^2mp^2}\lesssim\frac{(\operatorname{Var}N)^2}{\Psi},\qquad i\ge2, n 4 r − 2 − 2 i π ( G 2 , i , i + 1 ) ≲ n 2 m p 2 n 2 r − i π ( K 11 ⊕ i K 12 ) n 2 r − i π ( K 21 ⊕ i K 22 ) ≲ Ψ ( Var N ) 2 , i ≥ 2 , n 4 r − 1 − 2 i π ( G 3 , i , i + 1 ) ≲ n 2 r − i π ( K 11 ⊕ i K 12 ) n 2 r − i π ( K 21 ⊕ i K 22 ) n ≲ ( Var N ) 2 n , i ≥ 2 , n^{4r-1-2i}\pi(G_{3,i,i+1})\lesssim\frac{n^{2r-i}\pi(K_{11}\mathbin{\oplus_i}K_{12})n^{2r-i}\pi(K_{21}\mathbin{\oplus_i}K_{22})}{n}\lesssim\frac{(\operatorname{Var}N)^2}{n},\qquad i\ge2, n 4 r − 1 − 2 i π ( G 3 , i , i + 1 ) ≲ n n 2 r − i π ( K 11 ⊕ i K 12 ) n 2 r − i π ( K 21 ⊕ i K 22 ) ≲ n ( Var N ) 2 , i ≥ 2 , which gives us the required bound in this case.
No all-single intersections and m p 2 ≥ 1 mp^2 \ge1 m p 2 ≥ 1
In this subsection we deal with m p 2 ≥ 1 mp^2 \ge1 m p 2 ≥ 1 m p 3 ≤ c r mp^3 \le c_r m p 3 ≤ c r t , i , j t,i,j t , i , j § 6.4 and § 6.5 . We start with a general result.
Lemma 6.4. For t = 1 t=1 t = 1 j = i + 1 j=i+1 j = i + 1 t = 2 , 3 t=2,3 t = 2 , 3
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) ≲ ( 1 − p ) 2 \sum_{\mathbf I\subseteq\Gamma}(-1)^{|\mathbf I|}\mathbf P^{|\Gamma|-|\mathbf I|}\pi(G_{t,\mathbf I})\lesssim(1-p)^2 I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G t , I ) ≲ ( 1 − p ) 2 while if t = 1 t=1 t = 1 j ≥ i + 2 j\ge i+2 j ≥ i + 2
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G 1 , I ) ≲ 1 − p . \sum_{\mathbf I\subseteq\Gamma}(-1)^{|\mathbf I|}\mathbf P^{|\Gamma|-|\mathbf I|}\pi(G_{1,\mathbf I})\lesssim1-p. I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G 1 , I ) ≲ 1 − p . Proof. We will determine the values of π ( G t , I ) \pi(G_{t,\mathbf I}) π ( G t , I ) t t t I \mathbf I I ∣ I ∣ ≤ 1 |\mathbf I|\le1 ∣ I ∣ ≤ 1 ∣ I ∣ ≥ 2 |\mathbf I|\ge2 ∣ I ∣ ≥ 2 Lemma 4.21 . For I = Γ \mathbf I=\Gamma I = Γ Figure 2 )
π ( G t , Γ ) = π ( G t , { 11 , 12 } ) ∪ π ( G t , { 21 , 22 } ) π ( G t , { 11 , 12 } ) ∩ π ( G t , { 21 , 22 } ) + O ( m p 3 e − 2 m p 2 ) = P 4 π ( G t , { 11 , 12 } ) ∩ π ( G t , { 21 , 22 } ) π ( K 11 ∩ K 12 ) π ( K 21 ∩ K 22 ) + O ( m p 3 e − 2 m p 2 ) = O ( m p 3 e − 2 m p 2 ) + { P 4 π 2 ( K j ) π ( K j − i ) for t = 1 ; P 4 π 2 ( K i ) π 2 ( K j − i ) for t = 2 ; P 4 π 2 ( K i ) π ( K j − i ) for t = 3. \begin{aligned}
\pi(G_{t,\Gamma})=
\frac{\pi(G_{t,\{11,12\}})\cup\pi(G_{t,\{21,22\}})}
{\pi(G_{t,\{11,12\}})\cap\pi(G_{t,\{21,22\}})}
+O(mp^3e^{-2mp^2}) \\
&=\frac{\mathbf P^4}
{\pi(G_{t,\{11,12\}})\cap\pi(G_{t,\{21,22\}})\pi(\mathbf K_{11}\cap\mathbf K_{12})\pi(\mathbf K_{21}\cap\mathbf K_{22})}
+O(mp^3e^{-2mp^2}) \\
&=O(mp^3e^{-2mp^2})+
\begin{cases}
\dfrac{\mathbf P^4}{\pi^2(\mathbf K_j)\pi(\mathbf K_{j-i})} & \text{for }t=1;\\
\dfrac{\mathbf P^4}{\pi^2(\mathbf K_i)\pi^2(\mathbf K_{j-i})} & \text{for }t=2;\\
\dfrac{\mathbf P^4}{\pi^2(\mathbf K_i)\pi(\mathbf K_{j-i})} & \text{for }t=3.
\end{cases}
\end{aligned} π ( G t , Γ ) = π ( G t , { 11 , 12 } ) ∩ π ( G t , { 21 , 22 } ) π ( G t , { 11 , 12 } ) ∪ π ( G t , { 21 , 22 } ) + O ( m p 3 e − 2 m p 2 ) = π ( G t , { 11 , 12 } ) ∩ π ( G t , { 21 , 22 } ) π ( K 11 ∩ K 12 ) π ( K 21 ∩ K 22 ) P 4 + O ( m p 3 e − 2 m p 2 ) = O ( m p 3 e − 2 m p 2 ) + ⎩ ⎨ ⎧ π 2 ( K j ) π ( K j − i ) P 4 π 2 ( K i ) π 2 ( K j − i ) P 4 π 2 ( K i ) π ( K j − i ) P 4 for t = 1 ; for t = 2 ; for t = 3. We remark that in the case t = 1 t=1 t = 1 j = r j=r j = r K a 1 ∩ K a 2 ≅ K r \mathbf K_{a1}\cap\mathbf K_{a2}\cong\mathbf K_r K a 1 ∩ K a 2 ≅ K r a = 1 , 2 a=1,2 a = 1 , 2 i = 0 i=0 i = 0 O ( m p 3 e − 2 m p 2 ) O(mp^3e^{-2mp^2}) O ( m p 3 e − 2 m p 2 ) t = 1 t=1 t = 1 j = r j=r j = r
Similarly, we obtain for ∣ I ∣ = 3 |\mathbf I|=3 ∣ I ∣ = 3
π ( G t , I ) = O ( m p 3 e − 2 m p 2 ) + { P 3 π ( K j ) π ( K j − i ) for t = 1 ; P 3 π ( K i ) π ( K j − i ) for t = 2 ; P 3 π ( K i ) π ( K j − i ) for t = 3 and { 11 , 21 } ⊆ I ; P 3 π ( K i ) for t = 3 and { 12 , 22 } ⊆ I , \pi(G_{t,\mathbf I})=O(mp^3e^{-2mp^2})+
\begin{cases}
\dfrac{\mathbf P^3}{\pi(\mathbf K_j)\pi(\mathbf K_{j-i})} & \text{for }t=1;\\
\dfrac{\mathbf P^3}{\pi(\mathbf K_i)\pi(\mathbf K_{j-i})} & \text{for }t=2;\\
\dfrac{\mathbf P^3}{\pi(\mathbf K_i)\pi(\mathbf K_{j-i})} & \text{for }t=3\text{ and }\{11,21\}\subseteq\mathbf I;\\
\dfrac{\mathbf P^3}{\pi(\mathbf K_i)} & \text{for }t=3\text{ and }\{12,22\}\subseteq\mathbf I,
\end{cases} π ( G t , I ) = O ( m p 3 e − 2 m p 2 ) + ⎩ ⎨ ⎧ π ( K j ) π ( K j − i ) P 3 π ( K i ) π ( K j − i ) P 3 π ( K i ) π ( K j − i ) P 3 π ( K i ) P 3 for t = 1 ; for t = 2 ; for t = 3 and { 11 , 21 } ⊆ I ; for t = 3 and { 12 , 22 } ⊆ I , and for ∣ I ∣ = 2 |\mathbf I|=2 ∣ I ∣ = 2 x , y ∈ Γ x,y\in\Gamma x , y ∈ Γ
π ( G t , { x , y } ) = P 2 π ( K x ∩ K y ) + O ( m p 3 e − 2 m p 2 ) . \pi(G_{t,\{x,y\}})=\frac{\mathbf P^2}{\pi(\mathbf K_x\cap\mathbf K_y)}+O(mp^3e^{-2mp^2}). π ( G t , { x , y } ) = π ( K x ∩ K y ) P 2 + O ( m p 3 e − 2 m p 2 ) . In the sequel, we will consistently use the following relation, which follows from Fact 4.2. For any non-empty H H H m p 2 ≥ 1 mp^2\ge1 m p 2 ≥ 1
1 π ( H ) − 1 ≍ 1 − p . \frac{1}{\pi(H)}-1\asymp1-p. π ( H ) 1 − 1 ≍ 1 − p . Using all the above estimates, together with 1 − p ≲ e − m p 2 1-\mathbf{p}\lesssim e^{-mp^{2}} 1 − p ≲ e − m p 2 t = 1 t=1 t = 1 m p 2 ≥ 1 mp^{2}\geq1 m p 2 ≥ 1 m p 3 ≤ 1 mp^{3}\leq1 m p 3 ≤ 1
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G 1 , I ) = \sum_{I\subseteq\Gamma}(-1)^{|I|}\mathbf{P}^{|\Gamma|-|I|}\pi(G_{1,I}) = I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G 1 , I ) = P 4 ( 1 π 2 ( K j ) π ( K j − i ) − 4 π ( K j ) π ( K j − i ) + ( 2 π ( K j ) + 4 π ( K j − i ) ) − 4 + 1 ) + O ( m p 3 e − 2 m p 2 ) = P 4 [ 1 π ( K j − i ) ( 1 π ( K j ) − 1 ) 2 + ( 1 π ( K j − i ) − 1 ) ( 3 − 2 π ( K j ) ) ] + O ( m p 3 e − 2 m p 2 ) \begin{aligned}
\mathbf{P}^{4}\left(\frac{1}{\pi^{2}(K_j)\pi(K_{j-i})}-\frac{4}{\pi(K_j)\pi(K_{j-i})}+\left(\frac{2}{\pi(K_j)}+\frac{4}{\pi(K_{j-i})}\right)-4+1\right)+O(mp^{3}e^{-2mp^{2}}) \\
&=\mathbf{P}^{4}\left[\frac{1}{\pi(K_{j-i})}\left(\frac{1}{\pi(K_j)}-1\right)^{2}+\left(\frac{1}{\pi(K_{j-i})}-1\right)\left(3-\frac{2}{\pi(K_j)}\right)\right]+O(mp^{3}e^{-2mp^{2}})
\end{aligned} P 4 ( π 2 ( K j ) π ( K j − i ) 1 − π ( K j ) π ( K j − i ) 4 + ( π ( K j ) 2 + π ( K j − i ) 4 ) − 4 + 1 ) + O ( m p 3 e − 2 m p 2 ) = P 4 [ π ( K j − i ) 1 ( π ( K j ) 1 − 1 ) 2 + ( π ( K j − i ) 1 − 1 ) ( 3 − π ( K j ) 2 ) ] + O ( m p 3 e − 2 m p 2 ) ≲ ( 1 − p ) 2 + ( 1 − p ) + ( 1 − p ) 2 ≲ 1 − p . \lesssim(1-\mathbf{p})^{2}+(1-\mathbf{p})+(1-\mathbf{p})^{2}\lesssim1-\mathbf{p}. ≲ ( 1 − p ) 2 + ( 1 − p ) + ( 1 − p ) 2 ≲ 1 − p . In particular, this implies the latter bound from the assertion. Furthermore, for j = i + 1 j=i+1 j = i + 1 π ( K j − i ) = π ( N 1 ) = 1 \pi(K_{j-i})=\pi(N_{1})=1 π ( K j − i ) = π ( N 1 ) = 1
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G 1 , I ) = P 4 π ( K j − i ) ( 1 π ( K j ) − 1 ) 2 + O ( m p 3 e − 2 m p 2 ) ≲ ( 1 − p ) 2 , \sum_{I\subseteq\Gamma}(-1)^{|I|}\mathbf{P}^{|\Gamma|-|I|}\pi(G_{1,I})=\frac{\mathbf{P}^{4}}{\pi(K_{j-i})}\left(\frac{1}{\pi(K_j)}-1\right)^{2}+O(mp^{3}e^{-2mp^{2}})\lesssim(1-\mathbf{p})^{2}, I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G 1 , I ) = π ( K j − i ) P 4 ( π ( K j ) 1 − 1 ) 2 + O ( m p 3 e − 2 m p 2 ) ≲ ( 1 − p ) 2 , which gives us the first bound from the assertion in the case t = 1 t=1 t = 1 t = 2 t=2 t = 2
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G 2 , I ) = P 4 [ 1 π 2 ( K i ) π 2 ( K j − i ) − 4 π ( K i ) π ( K j − i ) + ( 2 + 2 π ( K i ) + 2 π ( K j − i ) ) − 4 + 1 ] + O ( m p 3 e − 2 m p 2 ) = P 4 [ ( 1 π ( K i ) π ( K j − i ) − 1 ) 2 − 2 ( 1 π ( K j − i ) − 1 ) ( 1 π ( K i ) − 1 ) ] + O ( m p 3 e − 2 m p 2 ) \begin{aligned}
\sum_{I\subseteq\Gamma}(-1)^{|I|}\mathbf{P}^{|\Gamma|-|I|}\pi(G_{2,I}) \\
&=\mathbf{P}^{4}\left[\frac{1}{\pi^{2}(K_i)\pi^{2}(K_{j-i})}-\frac{4}{\pi(K_i)\pi(K_{j-i})}+\left(2+\frac{2}{\pi(K_i)}+\frac{2}{\pi(K_{j-i})}\right)-4+1\right]
+O(mp^{3}e^{-2mp^{2}}) \\
&=\mathbf{P}^{4}\left[\left(\frac{1}{\pi(K_i)\pi(K_{j-i})}-1\right)^{2}-2\left(\frac{1}{\pi(K_{j-i})}-1\right)\left(\frac{1}{\pi(K_i)}-1\right)\right]+O(mp^{3}e^{-2mp^{2}})
\end{aligned} I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G 2 , I ) = P 4 [ π 2 ( K i ) π 2 ( K j − i ) 1 − π ( K i ) π ( K j − i ) 4 + ( 2 + π ( K i ) 2 + π ( K j − i ) 2 ) − 4 + 1 ] + O ( m p 3 e − 2 m p 2 ) = P 4 [ ( π ( K i ) π ( K j − i ) 1 − 1 ) 2 − 2 ( π ( K j − i ) 1 − 1 ) ( π ( K i ) 1 − 1 ) ] + O ( m p 3 e − 2 m p 2 ) ≲ ( 1 − p ) 2 . \lesssim(1-\mathbf{p})^{2}. ≲ ( 1 − p ) 2 . Eventually, for t = 3 t=3 t = 3
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G 3 , I ) = P 4 [ 1 π 2 ( K i ) π ( K j − i ) − ( 2 π ( K i ) π ( K j − i ) + 2 π ( K i ) ) + ( 2 π ( K i ) + 3 + 1 π ( K j − i ) ) − 4 + 1 ] + O ( m p 3 e − 2 m p 2 ) = P 4 1 π ( K j − i ) ( 1 π ( K i ) − 1 ) 2 + O ( m p 3 e − 2 m p 2 ) ≲ ( 1 − p ) 2 , \begin{aligned}
\sum_{I\subseteq\Gamma}(-1)^{|I|}\mathbf{P}^{|\Gamma|-|I|}\pi(G_{3,I}) \\
&=\mathbf{P}^{4}\left[\frac{1}{\pi^{2}(K_i)\pi(K_{j-i})}-\left(\frac{2}{\pi(K_i)\pi(K_{j-i})}+\frac{2}{\pi(K_i)}\right)+\left(\frac{2}{\pi(K_i)}+3+\frac{1}{\pi(K_{j-i})}\right)-4+1\right]
+O(mp^{3}e^{-2mp^{2}}) \\
&=\mathbf{P}^{4}\frac{1}{\pi(K_{j-i})}\left(\frac{1}{\pi(K_i)}-1\right)^{2}+O(mp^{3}e^{-2mp^{2}})\lesssim(1-\mathbf{p})^{2},
\end{aligned} I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G 3 , I ) = P 4 [ π 2 ( K i ) π ( K j − i ) 1 − ( π ( K i ) π ( K j − i ) 2 + π ( K i ) 2 ) + ( π ( K i ) 2 + 3 + π ( K j − i ) 1 ) − 4 + 1 ] + O ( m p 3 e − 2 m p 2 ) = P 4 π ( K j − i ) 1 ( π ( K i ) 1 − 1 ) 2 + O ( m p 3 e − 2 m p 2 ) ≲ ( 1 − p ) 2 , as required. □ \square □
The parameters considered in this section may be summarised in the following manner:
( α ) t = 1 , t = 2 , t = 3 , i ≥ 1 , i ≥ 1 , i ≥ 2 , j = i + 1 ≤ r − 1 , 3 ∨ ( i + 1 ) ≤ j ≤ r , i + 1 ≤ j ≤ r . \begin{aligned}
(\alpha)\quad&t=1, &&t=2, &&t=3,\\
&i\geq1, &&i\geq1, &&i\geq2,\\
&j=i+1\leq r-1, &&3\vee(i+1)\leq j\leq r, &&i+1\leq j\leq r.
\end{aligned} ( α ) t = 1 , i ≥ 1 , j = i + 1 ≤ r − 1 , t = 2 , i ≥ 1 , 3 ∨ ( i + 1 ) ≤ j ≤ r , t = 3 , i ≥ 2 , i + 1 ≤ j ≤ r . t = 1 , t = 1, t = 1 ,
( β ) (\beta) ( β )
i ≥ 0 , i \ge0, i ≥ 0 ,
i + 2 ≤ j ≤ r − 1. i + 2 \le j \le r - 1. i + 2 ≤ j ≤ r − 1.
Consider the case ( α ) (\alpha) ( α ) (49) we obtain
∣ V ( G 1 ) ∣ = 4 r − 2 j − ( j − i ) ≤ 4 r − 2 ⋅ 2 − 1 ≤ 4 r − 5 , ∣ V ( G 2 ) ∣ = 4 r − 2 j ≤ 4 r − 2 ⋅ 3 ≤ 4 r − 6 , ∣ V ( G 3 ) ∣ = 4 r − j − i ≤ 4 r − 3 − 2 ≤ 4 r − 5. \begin{aligned}
|V(G_1)| &= 4r - 2j - (j-i) \le4r - 2 \cdot2 - 1 \le4r - 5,\\
|V(G_2)| &= 4r - 2j \le4r - 2 \cdot3 \le4r - 6,\\
|V(G_3)| &= 4r - j - i \le4r - 3 - 2 \le4r - 5.
\end{aligned} ∣ V ( G 1 ) ∣ ∣ V ( G 2 ) ∣ ∣ V ( G 3 ) ∣ = 4 r − 2 j − ( j − i ) ≤ 4 r − 2 ⋅ 2 − 1 ≤ 4 r − 5 , = 4 r − 2 j ≤ 4 r − 2 ⋅ 3 ≤ 4 r − 6 , = 4 r − j − i ≤ 4 r − 3 − 2 ≤ 4 r − 5. Thus, Lemma 6.4 and (40) give us
n ∣ V ( G t ) ∣ ∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) ≲ n 4 r − 5 ( 1 − p ) 2 = ( n 2 r − 2 ( 1 − p ) ) 2 n ≲ ( Var [ X n ( r ) ] ) 2 n . n^{|V(G_t)|} \sum_{I\subseteq\Gamma}(-1)^{|I|}\mathbf{P}^{|\Gamma|-|I|}\pi(G_{t,I}) \lesssim n^{4r-5}(1-p)^2 = \frac{\left(n^{2r-2}(1-p)\right)^2}{n} \lesssim\frac{\left(\operatorname{Var}[X_n^{(r)}]\right)^2}{n}. n ∣ V ( G t ) ∣ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G t , I ) ≲ n 4 r − 5 ( 1 − p ) 2 = n ( n 2 r − 2 ( 1 − p ) ) 2 ≲ n ( Var [ X n ( r ) ] ) 2 . In the case ( β ) (\beta) ( β )
∣ V ( G 1 ) ∣ = 4 r − 3 j + i ≤ 4 r − 6 , |V(G_1)| = 4r - 3j + i \le4r - 6, ∣ V ( G 1 ) ∣ = 4 r − 3 j + i ≤ 4 r − 6 , and hence, by the latter part of Lemma 6.4,
n ∣ V ( G 1 ) ∣ ∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G 1 , I ) ≲ n 4 r − 6 ( 1 − p ) = ( n 2 r − 2 ( 1 − p ) ) 2 n 2 ( 1 − p ) ≲ ( Var [ X n ( r ) ] ) 2 ( 1 − p ) Ψ . n^{|V(G_1)|}\sum_{I\subseteq\Gamma}(-1)^{|I|}\mathbf{P}^{|\Gamma|-|I|}\pi(G_{1,I}) \lesssim n^{4r-6}(1-p) = \frac{\left(n^{2r-2}(1-p)\right)^2}{n^2(1-p)} \lesssim\frac{\left(\operatorname{Var}[X_n^{(r)}]\right)^2}{(1-p)\Psi}. n ∣ V ( G 1 ) ∣ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G 1 , I ) ≲ n 4 r − 6 ( 1 − p ) = n 2 ( 1 − p ) ( n 2 r − 2 ( 1 − p ) ) 2 ≲ ( 1 − p ) Ψ ( Var [ X n ( r ) ] ) 2 . All-single intersections
Here we consider the cases t = 1 t = 1 t = 1 i = 0 i = 0 i = 0 j = 1 j = 1 j = 1 t = 2 , 3 t = 2,3 t = 2 , 3 i = 1 i = 1 i = 1 j = 2 j = 2 j = 2 K a b K_{ab} K ab a b ∈ Γ ab \in\Gamma ab ∈ Γ p p p
For each I ⊆ Γ \mathbf{I} \subseteq\Gamma I ⊆ Γ t = 1 , 2 , 3 t = 1,2,3 t = 1 , 2 , 3 H t , I H_{t,\mathbf{I}} H t , I 4 r 4r 4 r G t , I G_{t,\mathbf{I}} G t , I ∣ Γ ∣ − ∣ I ∣ |\Gamma| - |\mathbf{I}| ∣Γ∣ − ∣ I ∣ r r r G t , I G_{t,\mathbf{I}} G t , I G t G_t G t K a b K_{ab} K ab a b ∈ Γ ∖ I ab \in\Gamma\setminus\mathbf{I} ab ∈ Γ ∖ I K a b K_{ab} K ab a b ∈ Γ ∖ I ab \in\Gamma\setminus\mathbf{I} ab ∈ Γ ∖ I G t , I G_{t,\mathbf{I}} G t , I H 1 , ∅ = K 11 ⊕ 0 K 12 ⊕ 0 K 21 ⊕ 0 K 22 H_{1,\varnothing}=K_{11}\oplus_0 K_{12}\oplus_0 K_{21}\oplus_0 K_{22} H 1 , ∅ = K 11 ⊕ 0 K 12 ⊕ 0 K 21 ⊕ 0 K 22 H 1 , { 11 , 12 } = G 1 , { 11 , 12 } ⊕ 0 K 21 ⊕ 0 K 22 ⊕ 0 N j H_{1,\{11,12\}}=G_{1,\{11,12\}}\oplus_0 K_{21}\oplus_0 K_{22}\oplus_0 N_j H 1 , { 11 , 12 } = G 1 , { 11 , 12 } ⊕ 0 K 21 ⊕ 0 K 22 ⊕ 0 N j G ( n , m , p ) \mathcal{G}(n,m,p) G ( n , m , p )
π ( H t , I ) = P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) . \pi(H_{t,\mathbf{I}})=\mathbf{P}^{|\Gamma|-|\mathbf{I}|}\pi(G_{t,\mathbf{I}}). π ( H t , I ) = P ∣Γ∣ − ∣ I ∣ π ( G t , I ) . Thus, the left-hand side of (47) may be rewritten as follows
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) = ∑ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H t , I ) . (54) \sum_{\mathbf{I}\subseteq\Gamma}(-1)^{|\mathbf{I}|}\mathbf{P}^{|\Gamma|-|\mathbf{I}|}\pi(G_{t,\mathbf{I}})
=
\sum_{\mathbf{I}\subseteq\Gamma}(-1)^{|\mathbf{I}|}\pi(H_{t,\mathbf{I}}). \tag*{(54)} I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G t , I ) = I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H t , I ) . (54) For each t = 1 , 2 , 3 t = 1,2,3 t = 1 , 2 , 3 I ⊆ Γ \mathbf{I}\subseteq\Gamma I ⊆ Γ { K a b } a b ∈ Γ \{K_{ab}\}_{ab\in\Gamma} { K ab } ab ∈ Γ H t , I H_{t,\mathbf{I}} H t , I C J ( H t , I ) C_{\mathbf{J}}(H_{t,\mathbf{I}}) C J ( H t , I ) J ⊆ Γ \mathbf{J}\subseteq\Gamma J ⊆ Γ H t , I H_{t,\mathbf{I}} H t , I
H t , I , J : = ( V ( H t , I ) , ⋃ a b ∈ J E ( K a b ) ) , H_{t,\mathbf{I},\mathbf{J}}:=\left(V(H_{t,\mathbf{I}}),\bigcup_{ab\in\mathbf{J}}E(K_{ab})\right), H t , I , J := ( V ( H t , I ) , ab ∈ J ⋃ E ( K ab ) ) , see (23) . Namely, C ∈ C J ( H t , I ) \mathcal{C} \in\mathcal{C}_{J}(H_{t,I}) C ∈ C J ( H t , I ) C = C J ∪ ⋅ ∪ ⋅ a b ∈ Γ ∖ J C a b \mathcal{C}=\mathcal{C}_{J}\mathbin{\mathop{\cup}\limits^{\cdot}}\mathop{\mathbin{\mathop{\cup}\limits^{\cdot}}}_{ab\in\Gamma\setminus J}\mathcal{C}_{ab} C = C J ∪ ⋅ ∪ ⋅ ab ∈ Γ ∖ J C ab C a b \mathcal{C}_{ab} C ab K a b K_{ab} K ab a b ∈ Γ ∖ J ab\in\Gamma\setminus J ab ∈ Γ ∖ J C J \mathcal{C}_{J} C J { K a b } a b ∈ J \{K_{ab}\}_{ab\in J} { K ab } ab ∈ J H t , I , J H_{t,I,J} H t , I , J C Γ ( H t , I ) \mathcal{C}_{\Gamma}(H_{t,I}) C Γ ( H t , I ) H t , I H_{t,I} H t , I { K a b } a b ∈ Γ \{K_{ab}\}_{ab\in\Gamma} { K ab } ab ∈ Γ
According to the definition, the families C J ( H t , I ) \mathcal{C}_{J}(H_{t,I}) C J ( H t , I ) J ⊆ Γ J\subseteq\Gamma J ⊆ Γ C ( H t , I ) = ∪ ⋅ J ⊆ Γ C J ( H t , I ) \mathcal{C}(H_{t,I})=\mathop{\mathbin{\mathop{\cup}\limits^{\cdot}}}_{J\subseteq\Gamma}\mathcal{C}_{J}(H_{t,I}) C ( H t , I ) = ∪ ⋅ J ⊆ Γ C J ( H t , I ) (54) , we obtain
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ P ∣ Γ ∣ − ∣ I ∣ π ( G t , I ) = ∑ I ⊆ Γ ( − 1 ) ∣ I ∣ ∑ J ⊆ Γ π ( H i , I , C J ( H i , I ) ) = ∑ ∅ ⊆ J ⊆ Γ ∑ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H i , I , C J ( H i , I ) ) . \begin{aligned}
\sum_{I\subseteq\Gamma}(-1)^{|I|}\mathop{\mathrm{P}}^{|\Gamma|-|I|}\pi(G_{t,I})=\sum_{I\subseteq\Gamma}(-1)^{|I|}\sum_{J\subseteq\Gamma}\pi(H_{i,I},\mathcal{C}_{J}(H_{i,I})) \\
&=\sum_{\varnothing\subseteq J\subseteq\Gamma}\sum_{I\subseteq\Gamma}(-1)^{|I|}\pi(H_{i,I},\mathcal{C}_{J}(H_{i,I})).
\end{aligned} I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ P ∣Γ∣ − ∣ I ∣ π ( G t , I ) = I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ J ⊆ Γ ∑ π ( H i , I , C J ( H i , I )) = ∅ ⊆ J ⊆ Γ ∑ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H i , I , C J ( H i , I )) . The assertion of Lemma 6.3 under current assumption is now a consequence of the following lemma. Namely, we show that all the inner sums above vanish if J ≠ Γ J\ne\Gamma J = Γ J = Γ J=\Gamma J = Γ
Lemma 6.5. For ∅ ⊆ J ⊊ Γ \varnothing\subseteq J\subsetneq\Gamma ∅ ⊆ J ⊊ Γ t = 1 , 2 , 3 t=1,2,3 t = 1 , 2 , 3
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H t , I , C J ( H t , I ) ) = 0. \sum_{I\subseteq\Gamma}(-1)^{|I|}\pi(H_{t,I},\mathcal{C}_{J}(H_{t,I}))=0. I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H t , I , C J ( H t , I )) = 0. Moreover, for m p ≥ 1 mp\geq1 m p ≥ 1 m p 3 ≤ c r mp^{3}\leq c_{r} m p 3 ≤ c r
n ∣ V ( G t ) ∣ ∣ ∑ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H t , I , C Γ ( H t , I ) ) ∣ ≲ ( Var N ) 2 n . n^{|V(G_{t})|}\left|\sum_{I\subseteq\Gamma}(-1)^{|I|}\pi(H_{t,I},\mathcal{C}_{\Gamma}(H_{t,I}))\right|\lesssim\frac{(\operatorname{Var}N)^{2}}{n}. n ∣ V ( G t ) ∣ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H t , I , C Γ ( H t , I )) ≲ n ( Var N ) 2 . In the proof of the lemma, we will take advantage of Lemma 4.20. For that, we first derive an auxiliary result that explains when the graphs under consideration are isomorphic.
Fact 6.1. Let I ′ , I ′ ′ , J ⊆ Γ I',I'',J\subseteq\Gamma I ′ , I ′′ , J ⊆ Γ I ′ ∩ J = I ′ ′ ∩ J I'\cap J=I''\cap J I ′ ∩ J = I ′′ ∩ J
H t , I ′ , J ≅ H t , I ′ ′ , J . H_{t,I',J}\cong H_{t,I'',J}. H t , I ′ , J ≅ H t , I ′′ , J . Proof. Recall that H t , I H_{t,I} H t , I G t , I G_{t,I} G t , I G t G_t G t I I I ∣ Γ ∣ − ∣ I ∣ |\Gamma|-|I| ∣Γ∣ − ∣ I ∣ H t , I , J H_{t,I,J} H t , I , J G t , I ∩ J G_{t,I\cap J} G t , I ∩ J ∣ J ∖ I ∣ |J\setminus I| ∣ J ∖ I ∣ I ′ ∩ J = I ′ ′ ∩ J I'\cap J=I''\cap J I ′ ∩ J = I ′′ ∩ J G t , I ′ ∩ J ≅ G t , I ′ ′ ∩ J G_{t,I'\cap J}\cong G_{t,I''\cap J} G t , I ′ ∩ J ≅ G t , I ′′ ∩ J ∣ J ∖ I ′ ∣ = ∣ J ∣ − ∣ J ∩ I ′ ∣ = ∣ J ∣ − ∣ J ∩ I ′ ′ ∣ = ∣ J ∖ I ′ ′ ∣ |J\setminus I'|=|J|-|J\cap I'|=|J|-|J\cap I''|=|J\setminus I''| ∣ J ∖ I ′ ∣ = ∣ J ∣ − ∣ J ∩ I ′ ∣ = ∣ J ∣ − ∣ J ∩ I ′′ ∣ = ∣ J ∖ I ′′ ∣ ∣ V ( H t , I ′ , J ) ∣ = ∣ V ( H t , I ′ ′ , J ) ∣ = 4 r |V(H_{t,I',J})|=|V(H_{t,I'',J})|=4r ∣ V ( H t , I ′ , J ) ∣ = ∣ V ( H t , I ′′ , J ) ∣ = 4 r □ \square □
Proof of Lemma 6.5. Let us rewrite the sum from the assertion by taking into consideration all the possible intersections of I I I J J J J ⊊ Γ J\subsetneq\Gamma J ⊊ Γ
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H t , I , C J ( H t , I ) ) = ∑ J ∗ ⊆ J ∑ I ⊆ Γ I ∩ J = J ∗ ( − 1 ) ∣ I ∣ π ( H t , I , C J ( H t , I ) ) = ∑ J ∗ ⊆ J ( − 1 ) ∣ J ∗ ∣ ∑ I ⊆ Γ I ∩ J = J ∗ ( − 1 ) ∣ I ∖ J ∣ π ( H t , I , C J ( H t , I ) ) . \begin{aligned}
\sum_{I\subseteq\Gamma}(-1)^{|I|}\pi(H_{t,I},\mathcal{C}_{J}(H_{t,I}))=\sum_{J^{*}\subseteq J}\sum_{\substack{I\subseteq\Gamma\\I\cap J=J^{*}}}(-1)^{|I|}\pi(H_{t,I},\mathcal{C}_{J}(H_{t,I})) \\
&=\sum_{J^{*}\subseteq J}(-1)^{|J^{*}|}\sum_{\substack{I\subseteq\Gamma\\I\cap J=J^{*}}}(-1)^{|I\setminus J|}\pi(H_{t,I},\mathcal{C}_{J}(H_{t,I})).
\end{aligned} I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H t , I , C J ( H t , I )) = J ∗ ⊆ J ∑ I ⊆ Γ I ∩ J = J ∗ ∑ ( − 1 ) ∣ I ∣ π ( H t , I , C J ( H t , I )) = J ∗ ⊆ J ∑ ( − 1 ) ∣ J ∗ ∣ I ⊆ Γ I ∩ J = J ∗ ∑ ( − 1 ) ∣ I ∖ J ∣ π ( H t , I , C J ( H t , I )) . Set J ∗ ⊆ J J^* \subseteq J J ∗ ⊆ J I ′ , I ′ ′ ⊂ Γ I',I'' \subset\Gamma I ′ , I ′′ ⊂ Γ I ′ ∩ J = I ′ ′ ∩ J = J ∗ I' \cap J = I'' \cap J = J^* I ′ ∩ J = I ′′ ∩ J = J ∗ { K a b } a b ∈ Γ \{K_{ab}\}_{ab\in\Gamma} { K ab } ab ∈ Γ H t , I ′ H_{t,I'} H t , I ′ H t , I ′ ′ H_{t,I''} H t , I ′′ Lemma 4.20 we get that
∀ I ′ , I ′ ′ , I ∩ I ′ π ( H t , I ′ , C J ( H t , I ′ ) ) = π ( H t , I ′ ′ , C J ( H t , I ′ ′ ) ) . \forall_{I',I'',I\cap I'}\pi(H_{t,I'},\mathcal{C}_J(H_{t,I'}))=\pi(H_{t,I''},\mathcal{C}_J(H_{t,I''})). ∀ I ′ , I ′′ , I ∩ I ′ π ( H t , I ′ , C J ( H t , I ′ )) = π ( H t , I ′′ , C J ( H t , I ′′ )) . Consequently,
∑ I ⊆ Γ I ∩ J = J ∗ ( − 1 ) ∣ I ∖ J ∣ π ( H t , I , C J ( H t , I ) ) = π ( H t , J ∗ , C J ( H t , J ∗ ) ) ∑ I ⊆ Γ I ∩ J = J ∗ ( − 1 ) ∣ I ∖ J ∣ = π ( H t , J ∗ , C J ( H t , J ∗ ) ) ∑ l = 0 ∣ Γ ∖ J ∣ ( ∣ Γ ∖ J ∣ l ) ( − 1 ) l = 0. \begin{aligned}
\sum_{\substack{I\subseteq\Gamma\\I\cap J=J^*}}(-1)^{|I\setminus J|}\pi(H_{t,I},\mathcal{C}_J(H_{t,I}))=\pi(H_{t,J^*},\mathcal{C}_J(H_{t,J^*}))\sum_{\substack{I\subseteq\Gamma\\I\cap J=J^*}}(-1)^{|I\setminus J|} \\
&=\pi(H_{t,J^*},\mathcal{C}_J(H_{t,J^*}))\sum_{l=0}^{|\Gamma\setminus J|}\binom{|\Gamma\setminus J|}{l}(-1)^l=0.
\end{aligned} I ⊆ Γ I ∩ J = J ∗ ∑ ( − 1 ) ∣ I ∖ J ∣ π ( H t , I , C J ( H t , I )) = π ( H t , J ∗ , C J ( H t , J ∗ )) I ⊆ Γ I ∩ J = J ∗ ∑ ( − 1 ) ∣ I ∖ J ∣ = π ( H t , J ∗ , C J ( H t , J ∗ )) l = 0 ∑ ∣Γ ∖ J ∣ ( l ∣Γ ∖ J ∣ ) ( − 1 ) l = 0. Consider now J = Γ J=\Gamma J = Γ C Γ ( H t , I ) \mathcal{C}_{\Gamma}(H_{t,I}) C Γ ( H t , I ) H t , I H_{t,I} H t , I { K a b } a b ∈ Γ \{K_{ab}\}_{ab\in\Gamma} { K ab } ab ∈ Γ
In the case m p 2 ≤ 1 mp^2\leq1 m p 2 ≤ 1 Lemma 4.25 . The graph F ( G t , ( K a b ) a b ∈ Γ ) F(G_t,(K_{ab})_{ab\in\Gamma}) F ( G t , ( K ab ) ab ∈ Γ ) Definition 4.24 ) is a star with 4 leaves for t = 1 t=1 t = 1 t = 2 t=2 t = 2 t = 3 t=3 t = 3 I ⊊ Γ I\subsetneq\Gamma I ⊊ Γ F ( H t , I , ( K a b ) a b ∈ Γ ) F(H_{t,I},(K_{ab})_{ab\in\Gamma}) F ( H t , I , ( K ab ) ab ∈ Γ ) F ( G t , ( K a b ) a b ∈ Γ ) F(G_t,(K_{ab})_{ab\in\Gamma}) F ( G t , ( K ab ) ab ∈ Γ ) t = 2 t=2 t = 2 I = Γ I=\Gamma I = Γ Lemma 4.25 and Theorem 2.1 , and observing that ∣ V ( G 1 ) ∣ = ∣ V ( G 3 ) ∣ = 4 r − 3 |V(G_1)|=|V(G_3)|=4r-3 ∣ V ( G 1 ) ∣ = ∣ V ( G 3 ) ∣ = 4 r − 3 ∣ V ( G 2 ) ∣ = 4 r − 4 |V(G_2)|=4r-4 ∣ V ( G 2 ) ∣ = 4 r − 4
n ∣ V ( G 1 ) ∣ ∑ I ⊆ Γ π ( H 1 , I , C Γ ( H 1 , I ) ) + n ∣ V ( G 3 ) ∣ ∑ I ⊆ Γ π ( H 3 , I , C Γ ( H 3 , I ) ) n^{|V(G_1)|}\sum_{I\subseteq\Gamma}\pi(H_{1,I},\mathcal{C}_{\Gamma}(H_{1,I}))+n^{|V(G_3)|}\sum_{I\subseteq\Gamma}\pi(H_{3,I},\mathcal{C}_{\Gamma}(H_{3,I})) n ∣ V ( G 1 ) ∣ I ⊆ Γ ∑ π ( H 1 , I , C Γ ( H 1 , I )) + n ∣ V ( G 3 ) ∣ I ⊆ Γ ∑ π ( H 3 , I , C Γ ( H 3 , I )) ≲ ( n r P ) 4 n 3 ( m p ) 2 = ( n r P ) 4 n m ( n 2 m p 2 ) ≲ 1 n m ( ( n r P ) 2 Ψ 2 ) 2 ≲ ( Var N ) 2 n , \lesssim\frac{(n^r\mathsf{P})^4}{n^3(mp)^2}=\frac{(n^r\mathsf{P})^4}{nm(n^2mp^2)}\lesssim\frac{1}{nm}\left(\frac{(n^r\mathsf{P})^2}{\Psi_2}\right)^2\lesssim\frac{(\operatorname{Var}N)^2}{n}, ≲ n 3 ( m p ) 2 ( n r P ) 4 = nm ( n 2 m p 2 ) ( n r P ) 4 ≲ nm 1 ( Ψ 2 ( n r P ) 2 ) 2 ≲ n ( Var N ) 2 , and
n ∣ V ( G 2 ) ∣ ∑ I ⊆ Γ π ( H 2 , I , C Γ ( H 2 , I ) ) n^{|V(G_2)|}\sum_{I\subseteq\Gamma}\pi(H_{2,I},\mathcal{C}_{\Gamma}(H_{2,I})) n ∣ V ( G 2 ) ∣ I ⊆ Γ ∑ π ( H 2 , I , C Γ ( H 2 , I )) ≲ ( n r P ) 4 n 4 ( m p ) 2 p = ( n r P ) 4 n ( n 2 m p 2 ) ( n m p ) ≲ 1 n ( n r P ) 2 Ψ 2 ( n r P ) 2 n m p ≲ ( Var N ) 2 n . \lesssim\frac{(n^r\mathsf{P})^4}{n^4(mp)^2p}=\frac{(n^r\mathsf{P})^4}{n(n^2mp^2)(nmp)}\lesssim\frac{1}{n}\frac{(n^r\mathsf{P})^2}{\Psi_2}\frac{(n^r\mathsf{P})^2}{nmp}\lesssim\frac{(\operatorname{Var}N)^2}{n}. ≲ n 4 ( m p ) 2 p ( n r P ) 4 = n ( n 2 m p 2 ) ( nm p ) ( n r P ) 4 ≲ n 1 Ψ 2 ( n r P ) 2 nm p ( n r P ) 2 ≲ n ( Var N ) 2 . Consider m p 2 ≥ 1 mp^2\geq1 m p 2 ≥ 1 E ( K a b ) E(K_{ab}) E ( K ab ) K a b K_{ab} K ab a b ∈ Γ ab\in\Gamma ab ∈ Γ K a b K_{ab} K ab a b ∈ Γ ab\in\Gamma ab ∈ Γ H t , I H_{t,I} H t , I t = 1 , 3 t=1,3 t = 1 , 3 t = 2 t=2 t = 2 K a b K_{ab} K ab a b ∈ Γ ab\in\Gamma ab ∈ Γ (12) and (40) , for every indecomposable clique cover C \mathcal{C} C H t , I H_{t,I} H t , I
n ∣ V ( G t , Γ ) ∣ π ( H t , I , C ) ≲ n 4 r − 3 ( m p 4 + ( m p 3 ) 2 ) e − 4 m p 2 ≲ ( n 2 r − 1 m p 3 e − 2 m p 2 ) 2 n ≲ ( Var N ) 2 n . n^{|V(G_{t,\Gamma})|}\pi(H_{t,I},\mathcal{C})\lesssim n^{4r-3}\left(mp^4+(mp^3)^2\right)e^{-4mp^2}\lesssim\frac{(n^{2r-1}mp^3e^{-2mp^2})^2}{n}\lesssim\frac{(\operatorname{Var}N)^2}{n}. n ∣ V ( G t , Γ ) ∣ π ( H t , I , C ) ≲ n 4 r − 3 ( m p 4 + ( m p 3 ) 2 ) e − 4 m p 2 ≲ n ( n 2 r − 1 m p 3 e − 2 m p 2 ) 2 ≲ n ( Var N ) 2 . Here, we used the inequality m p 4 ≤ ( m p 3 ) 2 mp^4\leq(mp^3)^2 m p 4 ≤ ( m p 3 ) 2 m p 2 ≥ 1 mp^2\geq1 m p 2 ≥ 1 (8) , this concludes the proof.
Case of G 3 , 1 , j G_{3,1,j} G 3 , 1 , j
First, let us explain why this case is not included in § 6.2.2 . Namely, the method employed there fails in this case, since π ( K 21 ⊕ 1 K 22 ) \pi(K_{21}\oplus_{1}K_{22}) π ( K 21 ⊕ 1 K 22 ) π ( K 21 ⊕ 1 K 22 ) − P 2 \pi(K_{21}\oplus_{1}K_{22})-P^{2} π ( K 21 ⊕ 1 K 22 ) − P 2
Let G 3 = G 3 , 1 , j G_{3}=G_{3,1,j} G 3 = G 3 , 1 , j j ≥ 3 j\geq3 j ≥ 3 (54) . The number of its terms is 2 ∣ Γ ∣ = 16 2^{|\Gamma|}=16 2 ∣Γ∣ = 16
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H 3 , I ) = ∑ I 1 ⊆ { 11 , 21 } ( − 1 ) ∣ I 1 ∣ ∑ I 2 ⊆ { 21 , 22 } ( − 1 ) ∣ I 2 ∣ π ( H 3 , I 1 ∪ I 2 ) . (55) \sum_{I\subseteq\Gamma}(-1)^{|I|}\pi(H_{3,I})=\sum_{I_{1}\subseteq\{11,21\}}(-1)^{|I_{1}|}\sum_{I_{2}\subseteq\{21,22\}}(-1)^{|I_{2}|}\pi(H_{3,I_{1}\cup I_{2}}). \tag*{(55)} I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H 3 , I ) = I 1 ⊆ { 11 , 21 } ∑ ( − 1 ) ∣ I 1 ∣ I 2 ⊆ { 21 , 22 } ∑ ( − 1 ) ∣ I 2 ∣ π ( H 3 , I 1 ∪ I 2 ) . (55) We observe that for F 0 = K r ⊕ 0 K r ⊕ 0 K r ⊕ 0 K r F_{0}=K_{r}\oplus_{0}K_{r}\oplus_{0}K_{r}\oplus_{0}K_{r} F 0 = K r ⊕ 0 K r ⊕ 0 K r ⊕ 0 K r F 1 = ( K r ⊕ 1 K r ) ⊕ 0 K r ⊕ 0 K r F_{1}=(K_{r}\oplus_{1}K_{r})\oplus_{0}K_{r}\oplus_{0}K_{r} F 1 = ( K r ⊕ 1 K r ) ⊕ 0 K r ⊕ 0 K r
I 1 = { 11 } , H 3 , { 11 } ∪ { 12 , 22 } ≅ H 3 , { 11 } ∪ { 12 } ≅ F 1 and H 3 , { 11 } ∪ { 22 } ≅ H 3 , { 11 } ∪ ∅ ≅ F 0 ; I 1 = { 21 } , H 3 , { 21 } ∪ { 12 , 22 } ≅ H 3 , { 21 } ∪ { 22 } ≅ F 1 and H 3 , { 21 } ∪ { 12 } ≅ H 3 , { 21 } ∪ ∅ ≅ F 0 ; I 1 = ∅ , H 3 , ∅ ∪ { 12 , 22 } ≅ H 3 , ∅ ∪ { 12 } ≅ H 3 , ∅ ∪ { 22 } ≅ H 3 , ∅ ∪ ∅ ≅ F 0 . \begin{aligned}
I_{1}&=\{11\},\qquad H_{3,\{11\}\cup\{12,22\}}\cong H_{3,\{11\}\cup\{12\}}\cong F_{1}\text{ and }H_{3,\{11\}\cup\{22\}}\cong H_{3,\{11\}\cup\varnothing}\cong F_{0};\\
I_{1}&=\{21\},\qquad H_{3,\{21\}\cup\{12,22\}}\cong H_{3,\{21\}\cup\{22\}}\cong F_{1}\text{ and }H_{3,\{21\}\cup\{12\}}\cong H_{3,\{21\}\cup\varnothing}\cong F_{0};\\
I_{1}&=\varnothing,\qquad H_{3,\varnothing\cup\{12,22\}}\cong H_{3,\varnothing\cup\{12\}}\cong H_{3,\varnothing\cup\{22\}}\cong H_{3,\varnothing\cup\varnothing}\cong F_{0}.
\end{aligned} I 1 I 1 I 1 = { 11 } , H 3 , { 11 } ∪ { 12 , 22 } ≅ H 3 , { 11 } ∪ { 12 } ≅ F 1 and H 3 , { 11 } ∪ { 22 } ≅ H 3 , { 11 } ∪ ∅ ≅ F 0 ; = { 21 } , H 3 , { 21 } ∪ { 12 , 22 } ≅ H 3 , { 21 } ∪ { 22 } ≅ F 1 and H 3 , { 21 } ∪ { 12 } ≅ H 3 , { 21 } ∪ ∅ ≅ F 0 ; = ∅ , H 3 , ∅ ∪ { 12 , 22 } ≅ H 3 , ∅ ∪ { 12 } ≅ H 3 , ∅ ∪ { 22 } ≅ H 3 , ∅ ∪ ∅ ≅ F 0 . Therefore, in (55) only the term with I 1 = { 11 , 21 } I_{1}=\{11,21\} I 1 = { 11 , 21 }
∑ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H 3 , I ) = ∑ I 2 ⊆ { 12 , 22 } ( − 1 ) ∣ I 2 ∣ π ( H 3 , { 11 , 21 } ∪ I 2 ) . (56) \sum_{I\subseteq\Gamma}(-1)^{|I|}\pi(H_{3,I})=\sum_{I_{2}\subseteq\{12,22\}}(-1)^{|I_{2}|}\pi(H_{3,\{11,21\}\cup I_{2}}). \tag*{(56)} I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H 3 , I ) = I 2 ⊆ { 12 , 22 } ∑ ( − 1 ) ∣ I 2 ∣ π ( H 3 , { 11 , 21 } ∪ I 2 ) . (56) Our aim now is to find the upper bound for (56) . We will achieve this using the same reasoning as in § 6.4 . For each H 3 , I H_{3,I} H 3 , I { 11 , 21 } ⊆ I ⊆ Γ \{11,21\}\subseteq I\subseteq\Gamma { 11 , 21 } ⊆ I ⊆ Γ { H 3 , I , i } i ∈ Λ \{H_{3,I,i}\}_{i\in\Lambda} { H 3 , I , i } i ∈ Λ Λ = { 0 , 1 , 2 } \Lambda=\{0,1,2\} Λ = { 0 , 1 , 2 } H 3 , I , 0 = G { 11 , 21 } = K 11 ⊕ j − 1 K 21 H_{3,I,0}=G_{\{11,21\}}=K_{11}\oplus_{j-1}K_{21} H 3 , I , 0 = G { 11 , 21 } = K 11 ⊕ j − 1 K 21 H 3 , I , 1 = K 12 H_{3,I,1}=K_{12} H 3 , I , 1 = K 12 H 3 , I , 2 = K 22 H_{3,I,2}=K_{22} H 3 , I , 2 = K 22 (23) , for L ⊆ Λ L\subseteq\Lambda L ⊆ Λ
H 3 , I , L = H 3 , I ∖ ⋃ i ∈ Λ ∖ L E ( H 3 , I , i ) = ( V ( H 3 , I ) , ⋃ i ∈ L E ( H 3 , I , i ) ) . H_{3,I,L}=H_{3,I}\setminus\bigcup_{i\in\Lambda\setminus L}E(H_{3,I,i})=\left(V(H_{3,I}),\bigcup_{i\in L}E(H_{3,I,i})\right). H 3 , I , L = H 3 , I ∖ i ∈ Λ ∖ L ⋃ E ( H 3 , I , i ) = ( V ( H 3 , I ) , i ∈ L ⋃ E ( H 3 , I , i ) ) . For L ⊆ Λ L\subseteq\Lambda L ⊆ Λ C L ( H 3 , I ) \mathcal{C}_{L}(H_{3,I}) C L ( H 3 , I ) H 3 , I , L H_{3,I,L} H 3 , I , L C L ( H 3 , I ) \mathcal{C}_{L}(H_{3,I}) C L ( H 3 , I ) L ⊆ Λ L\subseteq\Lambda L ⊆ Λ C ( H 3 , I ) \mathcal{C}(H_{3,I}) C ( H 3 , I ) I I I (9)
π ( H 3 , I ) = ∑ L ⊆ Λ π ( H 3 , I , C L ( H 3 , I ) ) , \pi(H_{3,I})=\sum_{L\subseteq\Lambda}\pi(H_{3,I},\mathcal{C}_{L}(H_{3,I})), π ( H 3 , I ) = L ⊆ Λ ∑ π ( H 3 , I , C L ( H 3 , I )) , which, applied to (56) , leads to
∑ { 11 , 21 } ⊆ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H 3 , I ) = ∑ L ⊆ { 0 , 1 , 2 } ∑ { 11 , 21 } ⊆ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H 3 , I , C L ( H 3 , I ) ) . (57) \sum_{\{11,21\}\subseteq I\subseteq\Gamma}(-1)^{|I|}\pi(H_{3,I})=\sum_{L\subseteq\{0,1,2\}}\sum_{\{11,21\}\subseteq I\subseteq\Gamma}(-1)^{|I|}\pi(H_{3,I},\mathcal{C}_{L}(H_{3,I})). \tag*{(57)} { 11 , 21 } ⊆ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H 3 , I ) = L ⊆ { 0 , 1 , 2 } ∑ { 11 , 21 } ⊆ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H 3 , I , C L ( H 3 , I )) . (57) Next, we are going to use Lemma 4.20 . For any J = L J=L J = L H J ( 1 ) = H 3 , I , L H_{J}^{(1)}=H_{3,I,L} H J ( 1 ) = H 3 , I , L H J ( 2 ) = H 3 , I ′ H_{J}^{(2)}=H_{3,I'} H J ( 2 ) = H 3 , I ′ I , I ′ ⊂ Γ I,I'\subset\Gamma I , I ′ ⊂ Γ (24) is naturally satisfied. Hence, H 3 , I , L ≅ H J ( 2 ) = H_{3,I,L}\cong H_{J}^{(2)}= H 3 , I , L ≅ H J ( 2 ) =
H 3 , I ′ , L H_{3,\mathbf{I}',\mathbf{L}} H 3 , I ′ , L π ( H 3 , I , C L ( H 3 , I ) ) = π ( H 3 , I ′ , C L ( H 3 , I ′ ) ) \pi(H_{3,\mathbf{I}},\mathcal{C}_{\mathbf{L}}(H_{3,\mathbf{I}}))=\pi(H_{3,\mathbf{I}'},\mathcal{C}_{\mathbf{L}}(H_{3,\mathbf{I}'})) π ( H 3 , I , C L ( H 3 , I )) = π ( H 3 , I ′ , C L ( H 3 , I ′ )) H 3 , I , L H_{3,\mathbf{I},\mathbf{L}} H 3 , I , L { 11 , 21 } ⊆ I ⊆ Γ \{11,21\}\subseteq\mathbf{I}\subseteq\Gamma { 11 , 21 } ⊆ I ⊆ Γ L ≠ Λ \mathbf{L}\ne\Lambda L = Λ Figure 10 )
Figure 10. In the first line are presented all graphs H 3 , I H_{3,\mathbf{I}} H 3 , I I \mathbf{I} I { 11 , 21 } ⊆ I \{11,21\}\subseteq\mathbf{I} { 11 , 21 } ⊆ I { H 3 , I , i } i ∈ Λ \{H_{3,\mathbf{I},i}\}_{i\in\Lambda} { H 3 , I , i } i ∈ Λ G I G_{\mathbf{I}} G I H 3 , I , L H_{3,\mathbf{I},\mathbf{L}} H 3 , I , L L = { 0 , 1 } \mathbf{L}=\{0,1\} L = { 0 , 1 }
H 3 , { 11 , 21 } , L ≅ H 3 , { 11 , 21 , 12 } , L ≅ H 3 , { 11 , 21 , 22 } , L ≅ H 3 , Γ , L for L = ∅ , { 0 } , { 1 , 2 } , { 1 } , { 2 } ; H_{3,\{11,21\},\mathbf{L}}\cong H_{3,\{11,21,12\},\mathbf{L}}\cong H_{3,\{11,21,22\},\mathbf{L}}\cong H_{3,\Gamma,\mathbf{L}}\qquad\text{for }\mathbf{L}=\varnothing,\{0\},\{1,2\},\{1\},\{2\}; H 3 , { 11 , 21 } , L ≅ H 3 , { 11 , 21 , 12 } , L ≅ H 3 , { 11 , 21 , 22 } , L ≅ H 3 , Γ , L for L = ∅ , { 0 } , { 1 , 2 } , { 1 } , { 2 } ; H 3 , { 11 , 21 } , L ≅ H 3 , { 11 , 21 , 22 } , L and H 3 , { 11 , 21 , 12 } , L ≅ H 3 , Γ , L for L = { 0 , 1 } ; H_{3,\{11,21\},\mathbf{L}}\cong H_{3,\{11,21,22\},\mathbf{L}}\text{ and }H_{3,\{11,21,12\},\mathbf{L}}\cong H_{3,\Gamma,\mathbf{L}}\qquad\text{for }\mathbf{L}=\{0,1\}; H 3 , { 11 , 21 } , L ≅ H 3 , { 11 , 21 , 22 } , L and H 3 , { 11 , 21 , 12 } , L ≅ H 3 , Γ , L for L = { 0 , 1 } ; H 3 , { 11 , 21 } , L ≅ H 3 , { 11 , 21 , 12 } , L and H 3 , { 11 , 21 , 21 } , L ≅ H 3 , Γ , L for L = { 0 , 2 } . H_{3,\{11,21\},\mathbf{L}}\cong H_{3,\{11,21,12\},\mathbf{L}}\text{ and }H_{3,\{11,21,21\},\mathbf{L}}\cong H_{3,\Gamma,\mathbf{L}}\qquad\text{for }\mathbf{L}=\{0,2\}. H 3 , { 11 , 21 } , L ≅ H 3 , { 11 , 21 , 12 } , L and H 3 , { 11 , 21 , 21 } , L ≅ H 3 , Γ , L for L = { 0 , 2 } . Consequently, for L = ∅ , { 0 } , { 1 , 2 } , { 1 } , { 2 } \mathbf{L}=\varnothing,\{0\},\{1,2\},\{1\},\{2\} L = ∅ , { 0 } , { 1 , 2 } , { 1 } , { 2 } (57) consists of terms that differ only by their sign, and it therefore vanishes. On the other hand, for L = { 0 , 1 } , { 0 , 2 } \mathbf{L}=\{0,1\},\{0,2\} L = { 0 , 1 } , { 0 , 2 } L ≠ Λ \mathbf{L}\ne\Lambda L = Λ (57) equals zero. Thus, we obtain
∑ { 11 , 21 } ⊆ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H 3 , I ) = ∑ { 11 , 21 } ⊆ I ⊆ Γ ( − 1 ) ∣ I ∣ π ( H 3 , I , C Λ ( H 3 , I ) ) . (58) \sum_{\{11,21\}\subseteq\mathbf{I}\subseteq\Gamma}(-1)^{|\mathbf{I}|}\pi(H_{3,\mathbf{I}})
=
\sum_{\{11,21\}\subseteq\mathbf{I}\subseteq\Gamma}(-1)^{|\mathbf{I}|}\pi(H_{3,\mathbf{I}},\mathcal{C}_{\Lambda}(H_{3,\mathbf{I}})). \tag*{(58)} { 11 , 21 } ⊆ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H 3 , I ) = { 11 , 21 } ⊆ I ⊆ Γ ∑ ( − 1 ) ∣ I ∣ π ( H 3 , I , C Λ ( H 3 , I )) . (58) Let m p ≥ 1 mp\ge1 m p ≥ 1 m p 2 ≤ 1 mp^{2}\le1 m p 2 ≤ 1 C Λ ( H 3 , I ) \mathcal{C}_{\Lambda}(H_{3,\mathbf{I}}) C Λ ( H 3 , I ) H 3 , I H_{3,\mathbf{I}} H 3 , I Lemma 4.25 (with partition { H 3 , I , i } i ∈ Λ \{H_{3,\mathbf{I},i}\}_{i\in\Lambda} { H 3 , I , i } i ∈ Λ F ( H 3 , I , { H 3 , I , i } i ∈ Λ ) F(H_{3,\mathbf{I}},\{H_{3,\mathbf{I},i}\}_{i\in\Lambda}) F ( H 3 , I , { H 3 , I , i } i ∈ Λ ) Definition 4.24 , is a forest), Lemma 5.1 (for π ( H 3 , I , 0 ) \pi(H_{3,\mathbf{I},0}) π ( H 3 , I , 0 ) H 3 , I , 0 ≅ K r ⊕ j − 1 K r H_{3,\mathbf{I},0}\cong K_r\oplus_{j-1}K_r H 3 , I , 0 ≅ K r ⊕ j − 1 K r j ≥ 3 j\ge3 j ≥ 3 (40) , we obtain for each { 11 , 21 } ⊆ I ⊆ Γ \{11,21\}\subseteq\mathbf{I}\subseteq\Gamma { 11 , 21 } ⊆ I ⊆ Γ
n 4 r − j − 1 π ( H 3 , I , C Λ ( H 3 , I ) ) ≲ n 4 r − j − 1 π ( H 3 , I , 0 ) P 2 ( m p ) 2 = n 2 r − ( j − 1 ) π ( K r ⊕ j − 1 K r ) ( n r P ) 2 ( n m p ) 2 n^{4r-j-1}\pi(H_{3,\mathbf{I}},\mathcal{C}_{\Lambda}(H_{3,\mathbf{I}}))
\lesssim
\frac{n^{4r-j-1}\pi(H_{3,\mathbf{I},0})P^{2}}{(mp)^{2}}
=
\frac{n^{2r-(j-1)}\pi(K_r\oplus_{j-1}K_r)(n^{r}P)^{2}}{(nmp)^{2}} n 4 r − j − 1 π ( H 3 , I , C Λ ( H 3 , I )) ≲ ( m p ) 2 n 4 r − j − 1 π ( H 3 , I , 0 ) P 2 = ( nm p ) 2 n 2 r − ( j − 1 ) π ( K r ⊕ j − 1 K r ) ( n r P ) 2 ≲ 1 n m p p ( n r P ) 2 Ψ ( n r P ) 2 n m p ≲ ( Var N ) 2 n m p ≤ ( Var N ) 2 n . \lesssim\frac{1}{n m p^p}\frac{(n^r\mathbf{P})^2}{\Psi}\frac{(n^r\mathbf{P})^2}{n m p} \lesssim\frac{(\operatorname{Var} N)^2}{n m p} \le\frac{(\operatorname{Var} N)^2}{n}. ≲ nm p p 1 Ψ ( n r P ) 2 nm p ( n r P ) 2 ≲ nm p ( Var N ) 2 ≤ n ( Var N ) 2 . This gives us the desired upper bound of (56) and concludes the proof in the case m p 2 ≤ 1 m p^2 \le1 m p 2 ≤ 1
Now let m p 2 ≥ 1 m p^2 \ge1 m p 2 ≥ 1 C Λ ( H 3 , I ) \mathcal{C}_{\Lambda}(H_{3,I}) C Λ ( H 3 , I ) H 3 , I H_{3,I} H 3 , I Lemma 4.23 (with partition { H ‾ i } i ∈ Λ \{\overline{H}_i\}_{i\in\Lambda} { H i } i ∈ Λ (40) give us for each { 11 , 21 } ⊆ I ⊆ Γ \{11,21\} \subseteq I \subseteq\Gamma { 11 , 21 } ⊆ I ⊆ Γ
n 4 r − 4 π ( H 3 , I , C Λ ( H 3 , I ) ) ≲ n 4 r − 4 m p 3 3 e − 3 m p 2 = ( n 2 r − 2 e − m p 2 ) ( n 2 r − 1 m p 3 e − 2 m p 2 ) n ≲ ( Var N ) 2 n . \begin{aligned}
n^{4r-4}\pi(H_{3,I},\mathcal{C}_{\Lambda}(H_{3,I})) \lesssim n^{4r-4}m p_3^3 e^{-3m p^2} \\
&= \frac{\left(n^{2r-2}e^{-m p^2}\right)\left(n^{2r-1}m p^3e^{-2m p^2}\right)}{n} \lesssim\frac{(\operatorname{Var} N)^2}{n}.
\end{aligned} n 4 r − 4 π ( H 3 , I , C Λ ( H 3 , I )) ≲ n 4 r − 4 m p 3 3 e − 3 m p 2 = n ( n 2 r − 2 e − m p 2 ) ( n 2 r − 1 m p 3 e − 2 m p 2 ) ≲ n ( Var N ) 2 . In view of (58) , this concludes the proof of Lemma 6.3 for i = 1 i=1 i = 1 j ≥ 3 j\ge3 j ≥ 3
Acknowledgements
Grzegorz Serafin’s work was supported by the National Science Centre (NCN), Poland, under research grant no. 2021/43/D/ST1/03244. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
