Longest cycles in certain bipartite graphs

Abstract

Let G be a connected bipartite graph with bipartition (X,Y) such that |X|≥|Y|(≥2), n=|X| and m=|Y|. Suppose, for all vertices x∈X and y∈Y, dist(x,y)=3 implies d(x)+d(y)≥n+1. Then G contains a cycle of length 2m. In particular, if m=n, then G is hamiltomian.