Torus-like graphs and their paired many-to-many disjoint path covers (original) (raw)

2021, Discrete Applied Mathematics

Given two disjoint vertex-sets, S = {s 1 ,. .. , s k } and T = {t 1 ,. .. , t k } in a graph, a paired many-to-many k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P 1 ,. .. , P k } that altogether cover every vertex of the graph, in which each path P i runs from s i to t i. 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.