  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 $p(n)\gg n^{-2/3}$ asymptotically almost surely has the
following resilience property: Every Hamiltonian subgraph $G'$ of $G(n,n,p)$
with more than $(1/2+o(1))n^2p$ 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.
