C7-DECOMPOSITIONS OF THE TENSOR PRODUCT OF COMPLETE GRAPHS

In this paper we consider a decomposition of K-m x K-n, where x denotes the tensor product of graphs, into cycles of length seven. We prove that for m, n >= 3, cycles of length seven decompose the graph K-m x K-n if and only if (1) either m or n is odd and (2) 14 | m(m - 1)n(n - 1). The results of this paper together with the results of [C-5-Decompositions of some regular graphs, Discrete Math. 306 (2006) 429451] and [C-5-Decompositions of the tensor product of complete graphs, Australasian J. Combinatorics 37 (2007) 285293], give necessary and sufficient conditions for the existence of a p-cycle decomposition, where p >= 5 is a prime number, of the graph K-m x K-n.