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

Pósa’s theorem states that any graph G whose degree sequence d₁ ≤ · · · ≤ d_n satisfies d_i ≥ 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 ≥ C log n and the i-th vertex degree (ordered increasingly) of G ⊆ 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.