Decomposition of Complete Bipartite Graphs

Abstract

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 $K_{d{2}^{d-2},d{2}^{d-1}}$ can be decomposed into copies of $Q_d$ for $d>1$.