Resilient degree sequences with respect to Hamilton cycles and matchings in random graphs

Abstract

Pósa's theorem states that any graph $G$ whose degree sequence $d_1\le \dots \le d_n$ satisfies $d_i\ge i+1$ for all $i<n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e. we prove a `resilience version' of Pósa's theorem: if $pn\ge C\log n$ and the $i$-th vertex degree (ordered increasingly) of $G\subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chvátal's theorem generalises Pósa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvátal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

To appear in the Electronic Journal of Combinatorics. This version corrects a couple of typos