On the Maximum Number of Independent Cycles in a Bipartite Graph

LetG=(V1, V2; E) be a bipartite graph with |V1|=|V2|=n⩾2k, wherekis a positive integer. Suppose that the minimum degree ofGis at leastk+1. We show that ifn>2k, thenGcontainskvertex-disjoint cycles. We also show that ifn=2k, thenGcontainsk−1 quadrilaterals and a path of order 4 such that all of them are vertex-disjoint. Moreover, the condition on degrees is sharp. We conjecture that whenn=2k,Gindeed containskvertex-disjoint quadrilaterals.