Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

Abstract

A paired many-to-many k-disjoint path cover (paired k-DPC for short) of a graph is a set of k vertex-disjoint paths joining k distinct source-sink pairs that altogether cover every vertex of the graph. We consider the problem of constructing paired 2-DPC’s in an m-dimensional bipartite HL-graph, Xm, and its application in finding the longest possible paths. It is proved that every Xm, m⩾4, has a fault-free paired 2-DPC if there are at most m−3 faulty edges and the set of sources and sinks is balanced in the sense that it contains the same number of vertices from each part of the bipartition. Furthermore, every Xm, m⩾4, has a paired 2-DPC in which the two paths have the same length if each source-sink pair is balanced. Using 2-DPC properties, we show that every Xm, m⩾3, with either at most m−2 faulty edges or one faulty vertex and at most m−3 faulty edges is strongly Hamiltonian-laceable.