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Pósa's theorem states that any graph G whose degree sequence d1 ≤ . . . ≤ dn satisfies di ≥ 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 'resilient' version of Pósa's theorem: if pn ≥ C log n and the i-th vertex degree (ordered increasingly) of G ⊆ Gn,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 resilient 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 resilient 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 Gn,p which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

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We study Hamiltonicity in random subgraphs of the hypercube Q n . Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of Q n according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree 2k, it contains k edge-disjoint Hamilton cycles, for any fixed k ∈ N. Secondly, we obtain a perturbation result: if H ⊆ Q n satisfies δ(H) ≥ αn with α > 0 fixed and we consider a random binomial subgraph Q n p of Q n with p ∈ (0, 1] fixed, then with high probability H ∪Q n p contains k edge-disjoint Hamilton cycles, for any fixed k ∈ N. In particular, both results resolve a long standing conjecture, posed e.g. by Bollobás, that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals 1/2. Our techniques also show that, with high probability, for all fixed p ∈ (0, 1] the graph Q n p contains an almost spanning cycle. Our methods involve branching processes, the Rödl nibble, and absorption.

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