Constructing edge-disjoint spanning trees in product networks

A Cartesian product network is obtained by applying the cross operation on two graphs. We study the problem of constructing the maximum number of edge-disjoint spanning trees (abbreviated to EDSTs) in Cartesian product networks. Let G=(V_G, E_G) be a graph having n₁ EDSTs and F=(V_F, E_F) 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=|E_G|·n₁(|V_G|-1) more edges than that are necessary for constructing n₁ EDSTs in it, and the graph F has t2=|E_F´-n₂(|V_F|-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.