Decompositions of Km, n and Kn into cubes

We consider the decomposition of the complete bipartite graph K'm,n into isomorphic copies of a d-cube. We present some general necessary conditions for such a decomposition and show that these conditions are sufficient for d 3 and d = 4. We also explore the d-cube decomposition of the complete graph Kn. Necessary and sufficient conditions for the existence of such a decomposition are known for d even and for d odd and n odd. We present a general strategy for constructing these decompositions for all values of d. We use this method to show that the necessary conditions are sufficient for d = 3.