C 10-decompositions of the tensor product of complete graphs

Abstract

In this paper, we consider C 10-decompositions of the tensor product of complete graphs, K m × K n , m, n ≥ 3. It is proved that the necessary conditions for such a decomposition are also sufficient. The cycle C 10 decomposes K m × K n if and only if either (1) n ≡ 0 or 6 (mod 10) and m is odd, (2) m ≡ 0 or 6 (mod 10) and n is odd, or (3) m or n ≡ 1 or 5 (mod 10). Using these conditions it can be shown that every even regular complete multipartite graph G is C 10-decomposable if the number of edges of G is divisible by 10.