Decomposition of Complete Bipartite Graphs

Conditions are given for decomposing \(K_{m,n}\) into edge-disjoint copies of a bipartite graph \(G\) by translating its vertices in the bipartition of the vertices of \(K_{m,n}\). A construction of the bipartite adjacency matrix of the \(d\)-cube \(Q_d\) is given leading to a convenient \(\alpha\)-valuation and a proof that \(K_{d2^{d-2},d2^{d-1}}\) can be decomposed into copies of \(Q_d\) for \(d > 1\).