Torus-like graphs and their paired many-to-many disjoint path covers

Abstract

Given two disjoint vertex-sets, S={s1,…,sk} and T={t1,…,tk} in a graph, a paired many-to-manyk-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 each path Pi runs from si to ti. In this paper, we propose a family of graphs, called torus-like graphs,that include torus networks, and reveal that a torus-like graph, if built from lower dimensional torus-like graphs that have good Hamiltonian and disjoint-path-cover properties, retain such good properties. As a result, every m-dimensional nonbipartite torus, m≥2, with at most f vertex and/or edge faults has a paired many-to-many k-disjoint path cover joining arbitrary disjoint sets S and T of size k each, subject to k≥2 and f+2k≤2m. The bound 2m on f+2k is nearly optimal.