Factorizations of the product of cycles

Abstract

An H-factorization of a graph G is a partition of the edge set of G into spanning subgraphs (or factors) each of whose components are isomorphic to a graph H. Let G be the Cartesian product of the cycles C1,C2,…,Cn with |Ci|=2ki≥4 for each i. El-Zanati and Eynden proved that G has a C-factorization, where C is a cycle of length s, if and only if s=2t with 2≤t≤k1+k2+⋯+kn. We extend this result to get factorizations of G into m-regular, m-connected and bipancyclic subgraphs. We prove that for 2≤m<2n, the graph G has an H-factorization, where H is an m-regular, m-connected and bipancyclic graph on s vertices, if and only if m divides 2n and s=2t with m≤t≤k1+k2+⋯+kn.