Constructing edge-disjoint spanning trees in product networks

Abstract

A Cartesian product network is obtained by applying the cross operation on two graphs. In this paper, we study the problem of constructing the maximum number of edge-disjoint spanning trees (abbreviated to EDSTs) in Cartesian product networks. Let G = (VG, EG) be a graph having n1 EDSTs and F = (VF, EF) be a graph having n2 EDSTs. Two methods are proposed for constructing EDSTs in the Cartesian product of G and F, denoted by G × F. The graph G has t1 = |EG| - n1(|VG| - 1) more edges than that are necessary for constructing n1 EDSTs in it, and the graph F has t2 = |EF| - n2(|VF| - 1) more edges than that are necessary for constructing n2 EDSTs in it. By assuming that t1 ≤ n1 and t2 ≤ n2, our first construction shows that n1 + n2 EDSTs can be constructed in G × F. Our second construction does not need any assumption and it constructs n1 + n2 - 1 EDSTs in G × F. By applying the proposed methods, it is easy to construct the maximum numbers of EDSTs in many important Cartesian product networks, such as hypercubes, tori, generalized hypercubes, mesh connected trees, and hyper Petersen networks.