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≥3 and every γ>0, there exists C=C(k,γ)>0 such that if Clog n/nk−1≤ p≤1, then asymptotically almost surely every spanning subhypergraph of H(k)(n,p) with increasing vertex-degree sequence g1≤⋯≤ gn satisfying gi≥((i/n)k−1+γ) binomn−1k−1p (1≤ 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.
Related
- Spectral extremal hypergraphs without long Berge cycles
- A Fan-Type Theorem for Hamiltonian Berge Cycles in 3-Uniform Hypergraphs
- Rainbow Berge Hamiltonicity in edge-colored random k-uniform hypergraphs
- Random independent sets in triangle-free and linear Berge-C4-free hypergraphs
- Resilience of rainbow Hamilton cycles in pseudorandom graphs
BibTeX
@misc{zhang2026pósatype,
title = {{Pósa-type resilience for Hamilton Berge cycles in random hypergraphs}},
author = {Liping Zhang and Ailian Chen and Qinghai Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08525},
eprint = {2611.08525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08525},
}
RIS
TY - UNPB
TI - Pósa-type resilience for Hamilton Berge cycles in random hypergraphs
AU - Liping Zhang
AU - Ailian Chen
AU - Qinghai Liu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.08525
UR - https://arxiv.org/abs/2611.08525
AB - We prove a degree-sequence resilience theorem for Hamilton Berge cycles in random uniform hypergraphs. For every fixed integer k≥3 and every γ>0, there exists C=C(k,γ)>0 such that if Clog n/n^k−1≤ p≤1, then asymptotically almost surely every spanning subhypergraph of H^(k)(n,p) with increasing vertex-degree sequence g_1≤⋯≤ g_n satisfying g_i≥((i/n)^k−1+γ) binomn−1k−1p (1≤ 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.
ER -