Pancyclic subgraphs of random graphs

An n-vertex graph is called pancyclic if it contains a cycle of length t for all 3 ≤ t ≤ n. In this paper, we study pancyclicity of random graphs in the context of resilience, and prove that if p n−1/2, then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than (12 +o(1)) ( n 2 ) p edges is pancyclic. This result is best possible in two ways. First, the range of p is asymptotically tight; second, the proportion 1 2 of edges cannot be reduced. Our theorem extends a classical theorem of Bondy, and is closely related to a recent work of Krivelevich, Lee, and Sudakov. The proof uses a recent result of Schacht (also independently obtained by Conlon and Gowers).