P\'osa'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 `resilience version'
of P\'osa's theorem: if pn≥Clogn 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 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 Gn,p which satisfies these conditions. This provides an
asymptotic characterisation of all degree sequences which resiliently guarantee
the existence of a perfect matching.Comment: To appear in the Electronic Journal of Combinatorics. This version
corrects a couple of typo