Partition of a bipartite graph into cycles

AbstractEl-Zahar (1984) conjectured that if G is a graph on n1+n2+⋯+nk vertices with ni⩾3 for 1⩽i⩽k a minimum degree δ(G)⩾⌈n1/2⌉+⌈n2/2⌉+⋯+⌈nk/2⌉, then G contains k vertex-disjoint cycles of lengths n1,n2,⋯,nk respectively. He proved this conjecture for k=2. In this note we consider a similar problem in bipartite graphs and prove: If G is a bipartite graph with partites V1 and V2 where |V1|=|V2|=n,n=n1+n2+⋯+nk for n1⩾n2⩾⋯⩾nk⩾2,k⩾2 and δ(G)⩾n1+n2+ ⋯+nk−1+nk/2, then G contains k vertex-disjoint cycles of lengths 2n1,2n2,…,2nk, respectively. We demonstrate by producing examples that the above result is best possible when k=2