CpCp-decompositions of some regular graphs

In this paper, for any prime p⩾11p⩾11, we consider CpCp-decompositions of Km×KnKm×Kn and Km*K¯n and also CpCp-factorizations of Km×KnKm×Kn, where ×× and ** denote the tensor product and wreath product of graphs, respectively, (Km*K¯n 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⩾3m,n⩾3, CpCp-decomposes Km×KnKm×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⩾3m⩾3, CpCp-decomposes Km*K¯n if and only if (1) (m-1)n(m-1)n is even and (2) p|m(m-1)n2. Except possibly for some valid pairs of integers m   and nn, the necessary conditions for the existence of CpCp-factorization of Km×KnKm×Kn are proved to be sufficient.