Unpaired many-to-many vertex-disjoint path covers of a class of bipartite graphs

Let W n denote any bipartite graph obtained by adding some edges to the n-dimensional hypercube Q n , and let S and T be any two sets of k vertices in different partite sets of W n . In this paper, we show that the graph W n has k vertex-disjoint ( S , T ) -paths containing all vertices of W n if and only if k = 2 n − 1 or the graph W n − ( S ∪ T ) has a perfect matching. Moreover, if the graph W n − ( S ∪ T ) has a perfect matching M, then the graph W n has k vertex-disjoint ( S , T ) -paths containing all vertices of W n and all edges in M. And some corollaries are given.