Unpaired many-to-many disjoint path covers in restricted hypercube-like graphs

Abstract

For two disjoint vertex-sets, S={s1,…,sk} and T={t1,…,tk} of a graph, an unpaired many-to-many k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P1,…,Pk} that altogether cover every vertex of the graph, in which Pi is a path from si to some tj for 1≤i,j≤k. A family of hypercube-like interconnection networks, called restricted hypercube-like graphs, includes most nonbipartite hypercube-like networks found in the literature, such as twisted cubes, crossed cubes, Möbius cubes, recursive circulant G(2m,4) of odd m, etc. In this paper, we show that every m-dimensional restricted hypercube-like graph, m≥5, with at most f faulty vertices and/or edges being removed has an unpaired many-to-many k-disjoint path cover joining arbitrary disjoint sets S and T of size k each subject to k≥2 and f+k≤m−1. The bound m−1 on f+k is the maximum possible.