C4p-frame of complete multipartite multigraphs

For two graphs G and H their wreath product has the vertex set in which two vertices (g (1), h (1)) and (g (2), h (2)) are adjacent whenever or g (1) = g (2) and . Clearly , where I (n) is an independent set on n vertices, is isomorphic to the complete m-partite graph in which each partite set has exactly n vertices. A subgraph of the complete multipartite graph containing vertices of all but one partite set is called partial factor. An H-frame of is a decomposition of into partial factors such that each component of it is isomorphic to H. In this paper, we investigate C (2k) -frames of , and give some necessary or sufficient conditions for such a frame to exist. In particular, we give a complete solution for the existence of a C (4p) -frame of , where p is a prime, as follows: For an integer m a parts per thousand yen 3 and a prime p, there exists a C (4p) -frame of if and only if and at least one of m, n must be even, when lambda is odd.