Paired 3-Disjoint Path Covers in Bipartite Torus-Like Graphs with Edge Faults

Abstract

Given two disjoint vertex-sets, [Formula: see text] and [Formula: see text] in a graph, a paired many-to-many [Formula: see text]-disjoint path cover joining [Formula: see text] and [Formula: see text] is a set of pairwise vertex-disjoint paths [Formula: see text] that altogether cover every vertex of the graph, in which each path [Formula: see text] runs from [Formula: see text] to [Formula: see text]. In this paper, we reveal that a bipartite torus-like graph, if built from lower dimensional torus-like graphs that have good disjoint-path-cover properties, retain such good property. As a result, an [Formula: see text]-dimensional bipartite torus, [Formula: see text], with at most [Formula: see text] edge faults has a paired many-to-many [Formula: see text]-disjoint path cover joining arbitrary disjoint sets [Formula: see text] and [Formula: see text] of size [Formula: see text] each such that [Formula: see text] contains the equal numbers of vertices from different parts of the bipartition.