Paired many-to-many disjoint path covers in faulty hypercubes

A paired many-to-many k-disjoint path cover (k-DPC for short) of a graph is a set of k disjoint paths joining k distinct source–sink pairs that cover all the vertices of the graph. Extending the notion of DPC, we define a paired many-to-many bipartite k-DPC of a bipartite graph G to be a set of k disjoint paths joining k distinct source–sink pairs that altogether cover the same number of vertices as the maximum number of vertices covered when the source–sink pairs are given in the complete bipartite, spanning supergraph of G. We show that every m  -dimensional hypercube, QmQm, under the condition that f or less faulty elements (vertices and/or edges) are removed, has a paired many-to-many bipartite k-DPC joining any k distinct source–sink pairs for any f   and k⩾1k⩾1 subject to f+2k⩽mf+2k⩽m. This implies that QmQm with m−2m−2 or less faulty elements is strongly Hamiltonian-laceable.