Almost Gorenstein edge rings of bipartite graphs and their $h$-vectors

Introduction

A Cohen–Macaulay graded ring is almost Gorenstein if it admits a degree-zero embedding into an appropriate shift of its canonical module whose cokernel is either zero or an Ulrich module. Here, an Ulrich module is a nonzero Cohen–Macaulay module whose multiplicity equals its minimal number of generators. Almost Gorenstein graded rings were introduced by Goto, Takahashi, and Taniguchi ([5]) and have since been studied in various settings, including graded rings arising from combinatorial objects (see, e.g., [14, 16, 17]).

By a theorem of Stanley, a Cohen–Macaulay standard graded domain is Gorenstein if and only if its $h$-vector is symmetric [21 Theorem 4.4]. This naturally raises the question of whether almost Gorensteinness is also determined by the $h$-vector. Higashitani investigated the relationship between almost Gorensteinness and $h$-vectors [10] and asked whether there exist two Cohen–Macaulay standard graded domains with the same $h$-vector such that one is almost Gorenstein and the other is not ([10, Problem 4.6]).

In this paper, we study almost Gorensteinness for edge rings of bipartite graphs, which form a class of normal Cohen–Macaulay domains. For a graph $G$, its edge ring $\Bbbk[G]$ is the $\Bbbk$-subalgebra of $\Bbbk[t_v : v \in V(G)]$ generated by $t_ut_v$ for $\{u,v\} \in E(G)$, with $\deg(t_ut_v)=1$. Almost Gorenstein edge rings have been studied for the following families:

• Complete multipartite graphs ([11]).

• Odd cycles sharing a common vertex ([12, 2]).

• Complete bipartite graphs with a matching deleted ([8]).

Our main result gives a complete characterization of bipartite graphs whose edge rings are almost Gorenstein. We also investigate the behavior of their $h$-vectors. To state our main theorem, we recall some graph-theoretic terminology (see § 2.2 for details). A graph is matching-covered if it is connected, has at least one edge, and every edge belongs to a perfect matching. For a vertex $v$ of $G$, let $N_G(v)$ denote its neighborhood. For a 2-connected matching-covered bipartite graph $G$, the filling construction introduced in [7] defines a set $\operatorname{Fill}(G)$ of missing edges whose addition yields a graph with a Gorenstein edge ring.

By the block decomposition and the tensor-product results of Matsushita and Miyashita ([15]), it suffices to consider $2$-connected bipartite graphs (see Proposition 2.9).

Our second main result concerns the symmetry of the $h$-vector. It applies to arbitrary bipartite graphs and shows that, in degree at least two, almost Gorensteinness forces symmetry except possibly between the linear and next-to-leading coefficients.

For a $2$-connected matching-covered bipartite graph $G$ with bipartition $X\sqcup Y$ and $|X|=|Y|=:n\geq3$, the $h$-polynomial has degree $s=n-1$. Let $\widehat{G}$ be the graph obtained from $G$ by adding all edges in $\operatorname{Fill}(G)$, and put $q:=|\operatorname{Fill}(G)|$. In this case, the proof gives the more precise identity

whenever $k[G]$ is almost Gorenstein (see (9)). Since $k[\widehat{G}]$ is Gorenstein ([7 Proposition 5.3]), this identity explains the coefficient symmetry in Theorem 1.2. This theorem gives an affirmative answer to [8 Question 4.8] when $s\geq2$.

As applications of Theorem 1.1, we recover known results on almost Gorenstein edge rings for two families of graphs (see Examples 3.4 and 3.5). Moreover, Example 3.6 gives two bipartite graphs whose edge rings have the common $h$-vector $(1,5,10,6,1)$, although only one of the rings is almost Gorenstein. Thus, almost Gorensteinness is not determined by the $h$-vector even among bipartite edge rings, answering [10 Problem 4.6] within this class.

This paper is organized as follows. In § 2, we recall basic facts on almost Gorenstein graded rings and bipartite edge rings. In Section 3, we prove our main theorems and present their applications.

Preliminaries

Throughout, $k$ is a field and all graphs are finite and simple.

Almost Gorenstein graded rings

We first recall some basic notions and invariants associated with standard graded Cohen–Macaulay domains (see, e.g., [3] for background). Let

be a standard graded Cohen–Macaulay domain with $R_0=k$ and $d:=\dim R$. Denote the unique homogeneous maximal ideal of $R$ by

and let $\omega_R$ be the canonical module of $R$. For a finitely generated graded $R$-module $M$, write $\mu(M)$ for its minimal number of homogeneous generators and $e(M)$ for its multiplicity with respect to $\mathfrak{m}$. Note that the inequality $e(M)\geq\mu(M)$ holds if $M$ is Cohen–Macaulay. Our convention for shifts is $M(b)_j:=M_{b+j}$. Put

The integers $a(R)$ and $r(R)$ are called the $a$-invariant and Cohen–Macaulay type of $R$, respectively. The Hilbert series of $R$ is the formal power series $\operatorname{Hilb}_{R}(t) := \sum_{k\geq0}(\dim_k R_k)t^k$. It can be written in the form

where $h_s\ne0$. We call the polynomial $h_0+h_1t+\cdots+h_st^s$ the $h$-polynomial of $R$, denote it by $h(R;t)$, and call the sequence $(h_0,h_1,\ldots,h_s)$ the $h$-vector of $R$. The index $s$ is called the socle degree of $R$. We have $s=d+a(R)$ and $h_s=\dim_k(\omega_R)_{-a(R)}$ (see [3 Section 4.4]). We call $R$ pseudo-Gorenstein if $h_s=1$.

Under our assumptions on $R$, there always exists a degree-preserving injection from $R$ to $\omega_R(-a(R))$ ([10 Proposition 2.2]), and its cokernel $C$ is Cohen–Macaulay of dimension $d-1$ if it is nonzero. The following proposition gives a numerical criterion for almost Gorensteinness:

Since the $h$-vector and Cohen–Macaulay type of a standard graded normal affine semigroup ring depend only on its semigroup, Proposition 2.2 shows that almost Gorensteinness is independent of the coefficient field. Thus, the following results, originally proved over an algebraically closed field of characteristic zero, hold over any field:

The following lemma will be used in the proof of our main theorem:

Proof. Extending scalars to $k(t)$ preserves the relevant dimensions, multiplicity, and number of generators, so we may assume that $k$ is infinite. Choose a linear system of parameters $\ell_1,\ldots,\ell_\delta$ for $M$ and put $Q=(\ell_1,\ldots,\ell_\delta)$. Since $M$ is Cohen–Macaulay and Ulrich,

The natural surjection $M/QM \longrightarrow M/\mathfrak{m}M$ is an isomorphism, so $\mathfrak{m}M=QM$. Moreover, $M$ is graded free over $k[\ell_1,\ldots,\ell_\delta]$. Since $M_j=0$ for $j\leq0$, the degree-one elements of a homogeneous free basis of $M$ form a $k$-basis of $M_1$. Thus, we have

Bipartite edge rings

We next fix the graph-theoretic notation and recall the definition and some properties of edge rings. For background on graph theory and edge rings, see [4], [22 Section 10], and [9 Section 5].

Let $G$ be a simple connected graph on vertex set $V(G)$ with edge set $E(G)$. For a vertex $v$, let $N_G(v)$ be its neighborhood and $d_G(v):=|N_G(v)|$. For $T\subset V(G)$, write $N_G(T):=\bigcup_{v\in T}N_G(v)$, let $G[T]$ be the induced subgraph, and set $G\setminus T:=G[V(G)\setminus T]$. We write $G\setminus v$ for $G\setminus\{v\}$. We say that $G$ is 2-connected if $|V(G)|\geq3$ and $G\setminus v$ is connected for any $v\in V(G)$.

In what follows, we assume that $G$ is bipartite with bipartition $V(G)=X\sqcup Y$. We identify pairs in $X\times Y$ with unordered pairs of vertices. For a subset $F\subset X\times Y$, let $G+F$ denote the graph on $V(G)$ with edge set $E(G)\cup F$. For $f\in X\times Y$, we write $G+f$ for $G+\{f\}$. For a nonempty set $T\subset X$, let $B(T)$ be the graph on $T\cup N_G(T)$ consisting of all edges incident with $T$. We call $T$ acceptable if $B(T)$ is connected and $G\setminus(T\cup N_G(T))$ is connected with at least one edge. The same terminology applies to subsets of $Y$.

For an unordered pair $e=\{u,v\}$ of distinct vertices, put $\rho(e)=e_u+e_v\in\mathbb{Z}^{V(G)}$, where $e_i$ denotes the unit vector indexed by $i$. Let $k[t_v:v\in V(G)]$ be the polynomial ring and let $t^U:=\prod_{u\in U}t_u$ for $U\subset V(G)$. The edge ring $k[G]$ and edge polytope $P_G$ of $G$ are defined as

Each edge monomial $t^e$ has degree one. The same definitions of the edge ring and edge polytope apply to graphs that are not bipartite. Set

If $G$ is connected and bipartite with at least one edge, then $k[G]=k[S_G]$ is normal ([18, 20]), hence Cohen–Macaulay ([13]), and

Normality gives $S_G=C_G\cap\mathbb{Z}S_G$, so the degree-$k$ monomials of $k[G]$ correspond bijectively to the lattice points in $kP_G$. Moreover, the canonical module of $k[G]$ is the monomial ideal with exponent vectors in $S_G\cap\operatorname{int}(C_G)$ ([3 Theorem 6.3.5]). Here, for a polytope or a cone $Q$, let $\operatorname{int}(Q)$ denote its relative interior.

Note that the form $L_T$ is non-negative on the cone $C_G$ for every $T\subset X$, even when $T$ is not acceptable.

A perfect matching is a set of pairwise disjoint edges covering all vertices. A graph is matching-covered if it is connected, has at least one edge, and every edge belongs to a perfect matching. The following theorem characterizes 2-connected matching-covered bipartite graphs in terms of neighborhood sizes and pseudo-Gorensteinness.

We next recall a construction that produces a Gorenstein edge ring by adding certain missing edges to a matching-covered graph. An acceptable set $T$ is tight if $|N_G(T)|-|T|=1$.

We refer to $\widehat{G}$ as the Gorenstein closure of $G$.

Let $G$ be a bipartite graph. An edge $e\in E(G)$ is called a bridge if removing $e$ increases the number of connected components of $G$. A block means a maximal 2-connected subgraph or a bridge. When $G$ has maximal 2-connected components $B_1,\ldots,B_m$ and bridges $e_1,\ldots,e_\ell$, we have

where $x_1,\ldots,x_\ell$ are indeterminates of degree one (see [22 Proposition 10.1.48]). Since each $B_i$ contains a cycle, $k[B_i]$ is not a polynomial ring, and hence $e(k[B_i])>1$. Thus, the following holds by [15 Corollaries 1.5 and 3.10]:

Therefore, it remains to characterize almost Gorenstein edge rings of 2-connected bipartite graphs.

Characterization and consequences

Key lemmas

We first establish two lemmas used in the proof of our main theorems.

Until the proof of Theorem 1.1, $G$ is a 2-connected matching-covered bipartite graph with $|X|=|Y|=:n$ and $n\ge3$. Set $R:=\mathbb{k}[G]$, $\mathfrak{m}:=\bigoplus_{k\ge1}R_k$, $q:=|\operatorname{Fill}(G)|$, and $\tau:=t^{V(G)}=\prod_{v\in V(G)}t_v$. Let $(h_0,\ldots,h_s)$ be the $h$-vector of $R$. Consider the exact sequence

Note that we have $C_j=0$ for $j<0$. Moreover, since $h_0=h_s=1$ by Theorem 2.6, the Hilbert-series formula in Proposition 2.2 yields $C_0=0$ and $\dim_{\mathbb{k}} C_1=h_{s-1}-h_1$.

By Proposition 2.8 (i), the classes $[\tau t^f]$ for $f\in\operatorname{Fill}(G)$ are nonzero and linearly independent in $C_1$. Since their number is $q=h_{s-1}-h_1=\dim_{\mathbb{k}} C_1$ by Proposition 2.8 (ii), they form a $\mathbb{k}$-basis of $C_1$. Consequently, $(\mathfrak{m}C)_2$ is spanned by the classes $[\tau t^f t^e]$ for $f\in\operatorname{Fill}(G)$ and $e\in E(G)$. The following lemma examines which of these classes are nonzero and distinct, and gives a lower bound for $\dim_{\mathbb{k}}(\mathfrak{m}C)_2$:

Proof. For $f=\{x,y\}\in\operatorname{Fill}(G)$ and $e=\{u,v\}\in E(G)$, with $u\in X$ and $v\in Y$, consider $[\tau t^f t^e]\in C_2$. This class vanishes precisely when $t^f t^e\in R_2$. The exponent vector of $t^f t^e$ is $\rho(f)+\rho(e)$, which has coordinate sum two on each side. There are only two possible pairings of the vertices $x,u$ with $y,v$, counted with multiplicity. Since $f$ is not an edge, the class vanishes exactly when $\{u,y\}$ and $\{x,v\}$ are both edges, that is, when $u\in P_f$ and $v\in Q_f$.

A nonzero class has at most two representations as $[\tau t^g t^{e'}]$ with $g\in\operatorname{Fill}(G)$ and $e'\in E(G)$. Besides $(f,e)$, the only possible representation uses one of the cross pairs $\{u,y\}$, $\{x,v\}$ as a filling edge and the other as an edge of $G$. If $e$ meets $f$, there is only one representation. Define

Nonzero monomial classes with distinct exponent vectors are linearly independent. The classes $[\tau t^f t^e]$ span $(\mathfrak{m}C)_2$, so

For fixed $f$, put

@page 6/12 starts_mid=0 ends_mid=1

Characterization and consequences

Key lemmas

We first establish two lemmas used in the proof of our main theorems.

Until the proof of Theorem 1.1, $G$ is a 2-connected matching-covered bipartite graph with $|X|=|Y|=:n$ and $n\ge3$. Set $R:=\mathbb{k}[G]$, $\mathfrak{m}:=\bigoplus_{k\ge1}R_k$, $q:=|\operatorname{Fill}(G)|$, and $\tau:=t^{V(G)}=\prod_{v\in V(G)}t_v$. Let $(h_0,\ldots,h_s)$ be the $h$-vector of $R$. Consider the exact sequence

Note that we have $C_j=0$ for $j<0$. Moreover, since $h_0=h_s=1$ by Theorem 2.6, the Hilbert-series formula in Proposition 2.2 yields $C_0=0$ and $\dim_{\mathbb{k}} C_1=h_{s-1}-h_1$.

By Proposition 2.8 (i), the classes $[\tau t^f]$ for $f\in\operatorname{Fill}(G)$ are nonzero and linearly independent in $C_1$. Since their number is $q=h_{s-1}-h_1=\dim_{\mathbb{k}} C_1$ by Proposition 2.8 (ii), they form a $\mathbb{k}$-basis of $C_1$. Consequently, $(\mathfrak{m}C)_2$ is spanned by the classes $[\tau t^f t^e]$ for $f\in\operatorname{Fill}(G)$ and $e\in E(G)$. The following lemma examines which of these classes are nonzero and distinct, and gives a lower bound for $\dim_{\mathbb{k}}(\mathfrak{m}C)_2$:

Proof. For $f=\{x,y\}\in\operatorname{Fill}(G)$ and $e=\{u,v\}\in E(G)$, with $u\in X$ and $v\in Y$, consider $[\tau t^f t^e]\in C_2$. This class vanishes precisely when $t^f t^e\in R_2$. The exponent vector of $t^f t^e$ is $\rho(f)+\rho(e)$, which has coordinate sum two on each side. There are only two possible pairings of the vertices $x,u$ with $y,v$, counted with multiplicity. Since $f$ is not an edge, the class vanishes exactly when $\{u,y\}$ and $\{x,v\}$ are both edges, that is, when $u\in P_f$ and $v\in Q_f$.

A nonzero class has at most two representations as $[\tau t^g t^{e'}]$ with $g\in\operatorname{Fill}(G)$ and $e'\in E(G)$. Besides $(f,e)$, the only possible representation uses one of the cross pairs $\{u,y\}$, $\{x,v\}$ as a filling edge and the other as an edge of $G$. If $e$ meets $f$, there is only one representation. Define

Nonzero monomial classes with distinct exponent vectors are linearly independent. The classes $[\tau t^f t^e]$ span $(\mathfrak{m}C)_2$, so

For fixed $f$, put

where pairs are identified with their unordered edges. The possible weights according to the location of $e$ are

location of $e$	$\lambda_{f,e}$
incident with $x$ or $y$	$1$
between $P_f$ and $Q_f$	$0$
between $U_f$ and $V_f$	$1$
$e \in B_f$	$1/2$ or $1$

The degree sum over $U_f \cup V_f$ counts each edge between $U_f$ and $V_f$ twice and each edge of $B_f$ once. Consequently,

The last equality uses $d_G(x) = n - 1 - |V_f|$ and $d_G(y) = n - 1 - |U_f|$. Summing over $f$ proves (3.2).

Since $G$ is 2-connected, every vertex has degree at least two. Moreover, every $f = \{x,y\} \in\operatorname{Fill}(G)$ satisfies

To see this, let $f = \{x,y\} \in\operatorname{Fill}(G)$. The non-negative form $L_{\{x\}}$ on $C_G$ is not identically zero, since connectedness gives an edge between $X\setminus\{x\}$ and $N_G(x)$. Since $1+\rho(f) \in\operatorname{int}(C_G)$, we have

The symmetric argument applies to $y$.

If the equality $\dim_k(\mathfrak{m}C)_2 = (2n-2)q$ holds, all terms $d_G(u)-2$ and $d_G(v)-2$ above vanish, and every edge of $B_f$ has weight $1/2$. Suppose $U_f \cup V_f \ne\varnothing$. Connectedness gives an edge between $U_f \cup V_f$ and its complement, so $B_f \ne\varnothing$. If $e = \{u,v\} \in B_f$ with $u \in U_f$ and $v \in Q_f$, its weight $1/2$ forces $\{u,y\} \in\operatorname{Fill}(G)$, since $\{x,v\} \in E(G)$. But an endpoint of a filling edge has degree at least three, whereas $d_G(u)=2$. For $u \in P_f$ and $v \in V_f$, the same argument uses $\{x,v\} \in\operatorname{Fill}(G)$ and contradicts $d_G(v)=2$. Thus $U_f=V_f=\varnothing$.

We say that $f=(x,y)\in(X\times Y)\setminus E(G)$ satisfies the neighborhood condition if $U_f=V_f=\varnothing$, equivalently, $N_G(x)=Y\setminus\{y\}$ and $N_G(y)=X\setminus\{x\}$. The following lemma describes how the Hilbert series changes when a missing edge satisfying the neighborhood condition is added to $G$:

Proof. Whenever a product of edge monomials of $G+f$ contains both $t^f$ and $t^{\{u,v\}}$, with $u\in X\setminus\{x\}$ and $v\in Y\setminus\{y\}$, we can replace their product using

The two monomials on the right correspond to edges of $G$ by the neighborhood condition. Each replacement reduces the number of factors $t^f$ by one. For a monomial outside $k[G]$, at least one such factor must remain. Repeating the replacement therefore expresses every such monomial in the form

The exponents $a_u$, $b_v$, and then $r$, are uniquely determined by the exponent vector. Conversely, for a monomial of degree $k$ in (8), the sum of its $x$- and $y$-exponents is $k+r>k$. Every degree-$k$ monomial of $k[G]$ has that sum at most $k$, since no edge of $G$ meets both $x$ and $y$. Thus all monomials in (8) lie outside $k[G]$. Their generating series is

Proofs of the main results

We now prove Theorem 1.1. The proof also gives the numerical formulas recorded in Corollary 3.3.

Proof of Theorem 1.1. Put $R:=k[G]$ and let $s$ be its socle degree. Since $G$ is 2-connected, it has a cycle and $s\geq1$. The case $s=1$ occurs exactly when $G=K_{2,m}$ for some $m\geq2$; this follows from [19 Lemma 4.2 and Theorem 4.6 and its proof], since $G$ is 2-connected. By Theorem 2.3 (i), these rings are almost Gorenstein. In what follows, we assume $s\geq2$.

Assume first that $R$ is almost Gorenstein. By Theorem 2.3 (ii), we have $h_s=1$, so $G$ is matching-covered by Theorem 2.6. Put $n:=|X|=|Y|$ and $q:=|\operatorname{Fill}(G)|$. Then (2.2) and Theorem 2.6 give $\dim R=2n-1$, $a(R)=-n$, and $s=n-1$. In particular, $n=s+1\geq3$.

If $\operatorname{Fill}(G)=\varnothing$, equivalently, $k[G]$ is Gorenstein, the neighborhood condition holds automatically. Otherwise the cokernel $C$ in (3.1) is nonzero and Ulrich by Proposition 2.2, and Lemma 2.4 gives $\dim_k(\mathfrak{m}C)_2=(\dim k[G]-1)\cdot\dim_k C_1=(2n-2)q$. By Lemma 3.1, $U_f=V_f=\varnothing$ for every $f\in\operatorname{Fill}(G)$.

Conversely, suppose that (ii) holds, that is, $G$ is matching-covered and every $f\in\operatorname{Fill}(G)$ satisfies the neighborhood condition. Two different edges of $\operatorname{Fill}(G)$ cannot share an endpoint, since each such endpoint has exactly one non-neighbor across the bipartition. Thus $\operatorname{Fill}(G)$ is a matching in $\widehat{G}$. Adjoin its edges one at a time. For every remaining filling edge its endpoint neighborhoods are unchanged, so Lemma 3.2 applies at each step and yields

The ring $k[\widehat{G}]$ is Gorenstein by Proposition 2.8 (iii), so its $h$-polynomial is palindromic of degree $s=n-1$. If $q=0$, then $R$ is Gorenstein and hence almost Gorenstein. If $q>0$, (7) and the neighborhood condition give $n\geq4$. The palindromicity of $h(k[\widehat{G}];t)=h(R;t)+qt$ and Proposition 2.2 imply

For each $f=\{x,y\}\in\operatorname{Fill}(G)$ and $1\leq j\leq n-3$, the exponent vector

lies in $\operatorname{int}(C_G)$. Indeed, its coordinates are positive. For an acceptable $T\subset X$, the only way that the added coordinates decrease $L_T$ is $x\in T$ and $y\notin N_G(T)$. The neighborhood condition then forces $T=\{x\}$, and

For every other acceptable set, the value is at least $|N_G(T)|-|T|\ge1$. Normality shows that these lattice vectors give canonical monomials.

All these monomials are minimal generators of $\omega_R$. If subtracting an edge vector left an interior vector, positivity of every coordinate would force both endpoints of the subtracted edge to lie in $\{x,y\}$. But $f$ is not an edge of $G$. The monomial $\tau$ is also a minimal generator of $\omega_R$. These monomials have distinct exponent vectors, so

On the other hand, Proposition 2.2 gives

Equality follows, proving almost Gorensteinness. □

The proof of Theorem 1.1 also yields our second main result:

This gives the coefficient symmetry asked for in [8 Question 4.8] when $s\ge2$. The degree restriction is necessary: $k[K_{2,m}]$ is almost Gorenstein with $h$-polynomial $1+(m-1)t$, which has $h_0\ne h_s$ when $m\ge3$.

Examples

We first recover the almost Gorenstein criteria for two known families.

The next example uses a bipartite realization of an edge ring of a non-bipartite graph.

The last example shows that almost Gorensteinness is not determined by the $h$-vector, even among edge rings of 2-connected matching-covered bipartite graphs.

Example 3.6. Let $X=\{x_1,\ldots,x_5\}$ and $Y=\{y_1,\ldots,y_5\}$. Define bipartite graphs $G$ and $G'$ with bipartition $X\sqcup Y$ by the following neighborhoods; each entry lists the indices of the $y$-vertices.

	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$
$G$	$\{2,3,4,5\}$	$\{1,2,3\}$	$\{1,2,3\}$	$\{1,4\}$	$\{1,5\}$
$G'$	$\{3,4,5\}$	$\{1,2,3\}$	$\{1,2,3\}$	$\{1,4,5\}$	$\{1,5\}$

Thus $G'$ is obtained from $G$ by deleting $\{x_1,y_2\}$ and adding $\{x_4,y_5\}$. The two graphs are drawn in Figure 2.

Using Macaulay2 ([6]), we obtain

Both graphs are 2-connected. Since $h_{4}=1$, both graphs are matching-covered by Theorem 2.6. By Proposition 2.8 (ii), $|\operatorname{Fill}(G)|=|\operatorname{Fill}(G')|=h_{3}-h_{1}=1$. Since every vertex of $X \setminus\{x_{1}\}$ is adjacent to $y_{1}$ and $\{x_{1}\}$ is not tight in either graph, Definition 2.7 shows that $\{x_{1},y_{1}\}$ belongs to both filling sets. Hence

For $G$, the neighborhoods of $x_{1}$ and $y_{1}$ are $Y \setminus\{y_{1}\}$ and $X \setminus\{x_{1}\}$, respectively. For $G'$, $N_{G'}(x_{1})=\{y_{3},y_{4},y_{5}\}$, so the required neighborhood equality fails. Thus, by Theorem 1.1, $k[G]$ is almost Gorenstein but $k[G']$ is not.

The two rings in Example 3.6 are normal Cohen–Macaulay homogeneous domains. They therefore answer [10 Problem 4.6] even when both domains are required to be bipartite edge rings. Since their common $h$-polynomial becomes palindromic after adding $t$, the palindromicity condition in Corollary 3.3 does not imply almost Gorensteinness.

Acknowledgement

The second author was partially supported by Grant-in-Aid for JSPS Fellows Grant JP25KJ0047. The authors used OpenAI’s ChatGPT and Codex to assist with drafting and editing the manuscript and to discuss some mathematical arguments. The authors take full responsibility for the content of this paper. (Y. Hatasa) INDEPENDENT RESEARCHER, JAPAN Email address: [email protected]

(K. Matsushita) GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, THE UNIVERSITY OF TOKYO, KOMABA, MEGURO-KU, TOKYO 153-8914, JAPAN Email address: [email protected]