Cube factorizations of complete multipartite graphs

Let lambda K(hu) denote the lambda-fold complete multipartite graph with u parts of size h. A cube factorization of lambda K(hu) is a uniform 3-factorization of lambda K(hu) in which the components of each factor are cubes. We show that there exists a cube factorization of lambda K(hu) if and only if uh equivalent to 0 (mod 8), lambda(u - 1)h equivalent to 0 (mod 3) and u >= 2. It gives a new family of uniform 3-factorizations of lambda K(hu). We also establish the necessary and sufficient conditions for the existence of cube frames of lambda K(hu).