Paired Many-to-Many 3-Disjoint Path Covers in Bipartite Toroidal Grids

Abstract

Given two disjoint vertex-sets, S = {s1,. .. , s k } and T = {t1,. .. , 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 {P1,. .. , P k } that altogether cover every vertex of the graph, in which each path Pi runs from si to ti. In this paper, we first study the disjoint-path-cover properties of a bipartite cylindrical grid. Based on the findings, we prove that every bipartite toroidal grid, excluding the smallest one, has a paired many-to-many 3-disjoint path cover joining S = {s1, s2, s3} and T = {t1, t2, t3} if and only if the set S ∪ T contains the equal numbers of vertices from different parts of the bipartition.