Pósa-type resilience for Hamilton Berge cycles in random hypergraphs

Abstract

We prove a degree-sequence resilience theorem for Hamilton Berge cycles in random uniform hypergraphs. For every fixed integer $k\ge3$ and every $γ>0$, there exists $C=C(k,γ)>0$ such that if $C\log n/n^{k-1}\le p\le1$, then asymptotically almost surely every spanning subhypergraph of $H^{(k)}(n,p)$ with increasing vertex-degree sequence $g_1\le\cdots\le g_n$ satisfying $g_i\ge\left(\left(\frac{i}{n}\right)^{k-1}+γ\right) \binom{n-1}{k-1}p \qquad(1\le i<n/2)$ contains a Hamilton Berge cycle. The term $(i/n)^{k-1}$ agrees with the normalized asymptotic degree bound in Salia's deterministic Pósa-type theorem. Our result gives a hypergraph analogue of the random-graph theorem of Condon et al. and strengthens the sharp Dirac-type resilience theorem of Chen and Zhang at the same logarithmic probability order. The deletion-resilience formulation has optimal probability order up to constant factors. The coefficient $1$ of $(i/n)^{k-1}$ cannot be replaced by a smaller constant. The proof controls adaptive rotations and representative-edge incidences uniformly. It also gives an endpoint-growth estimate under nonuniform deletion bounds.