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

P'osa’s theorem states that any graph $G$ whose degree sequence $d_1 \le \ldots \le d_n$satisfies$d_i \ge i+1$forall$i < n/2$ 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 P'osa’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 $i<n\mathrm{/}2$, 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. Chv'atal’s theorem generalises P'osa’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 Chv'atal’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.