Graph Decompositions into Generalized Cubes

For a positive integer \(d\), the usual \(d\)-dimensional cube \(Q_d\) is defined to be the graph \((K_2)^d\), the Cartesian product of \(d\) copies of \(K_2\). We define the generalized cube \(Q_{d,k}\) to be the graph \((K_k)^d\) for positive integers \(d\) and \(k\). We investigate the decompositions of the complete graph \(K_{k^d}\) and the complete \(k\)-partite graph \(K_{k \times k^{d-1}}\) into generalized cubes when \(k\) is the power of a prime and \(d\) is any positive integer, and some generalizations. We also use these results to show that \(Q_{5}\) divides \(K_{96}\).