Cp-decompositions of some regular graphs

In this paper, for any prime p 11, we consider Cp-decompositions of Km × Kn and Km ∗ Kn and also Cp-factorizations of Km × Kn, where × and ∗ denote the tensor product and wreath product of graphs, respectively, (Km ∗ Kn is isomorphic to the complete m-partite graph in which each partite set has exactly n vertices). It has been proved that for m, n 3, Cp-decomposes Km ×Kn if and only if (1) either m or n is odd and (2) p |mn(m− 1)(n− 1). Further, it is shown that for m 3, Cp-decomposes Km ∗Kn if and only if (1) (m− 1)n is even and (2) p |m(m− 1)n2. Except possibly for some valid pairs of integers m and n, the necessary conditions for the existence of Cp-factorization of Km ×Kn are proved to be sufficient. © 2005 Published by Elsevier B.V.