Abstract
In this paper we prove an optimal co-degrees resilience property for the binomial $k$-uniform hypergraph model $H_{n,p}^k$ with respect to perfect matchings. That is, for a sufficiently large $n$ which is divisible by $k$, and $p\geq C_k\log {n}/n$, we prove that with high probability every subgraph $H\subseteq H^k_{n,p}$ with minimum co-degree (meaning, the number of supersets every set of size $k-1$ is contained in) at least $(1/2+o(1))np$ contains a perfect matching.
Highlights
A perfect matching in a k-uniform hypergraph H is a collection of vertex-disjoint edges, covering every vertex of V (H) exactly once
A famous result by Dirac [2] asserts that every graph G on n vertices and with minimum degree δ(G) n/2 contains a Hamiltonian cycle
Extending this result to hypergraphs provides us with some interesting cases, as one can
Summary
A perfect matching in a k-uniform hypergraph H is a collection of vertex-disjoint edges, covering every vertex of V (H) exactly once. Following a long line of work in studying this property, which is expanded upon in the former surv√ey, Kuhn and Osthus proved in [6] that every k-uniform hypergraph with δk−1 n/2 + 2n√log n contains a perfect matching. After a few decades of study, in 2008 Johansson, Kahn and Vu [3] managed to determine the threshold Among their results, one of particular note is that for p C log n/nk−1, whp Hnk,p contains a perfect matching. Note that if p = o(log n/n) whp there exists a (k − 1)-set of vertices which is not contained in any edge and for the study of (k − 1)-resilience, it is natural to restrict our attention to p C log n/n (which is significantly above the threshold for a perfect matching as obtained in [3]). Whp the resulting subhypergraph has δk−1(H) (1/2 − ε)np and does not contain a perfect matching
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