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, X-m, and its application in finding the longest possible paths. It is proved that every X-m, 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 X-m, 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 X-m, 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. (C) 2013 Elsevier Inc. All rights reserved.