Bipancyclic subgraphs in random bipartite graphs

A bipartite graph on 2n vertices is bipancyclic if it contains cycles of all even lengths from 4 to 2n. In this paper we prove that the random bipartite graph G(n, n, p) with p(n) À n−2/3 asymptotically almost surely has the following resilience property: Every Hamiltonian subgraph G′ of G(n, n, p) with more than (1/2 + o(1))n2p edges is bipancyclic. This result is tight in two ways. First, the range of p is essentially best possible. Second, the proportion 1/2 of edges cannot be reduced. Our result extends a classical theorem of Mitchem and Schmeichel.