Cube Factorizations of Complete Multipartite Graphs

Abstract

Let \(\lambda K_{h^u}\) denote the \(\lambda\)-fold complete multipartite graph with \(u\) parts of size \(h\). A cube factorization of \(\lambda K_{h^u}\) is a uniform \(3\)-factorization of \(\lambda K_{h^u}\) in which the components of each factor are cubes. We show that there exists a cube factorization of \(\lambda K_{h^u}\) if and only if \(uh \equiv 0 \pmod{8}\), \(\lambda (u-1)h \equiv 0 \pmod{3}\), and \(u \geq 2\). It gives a new family of uniform \(3\)-factorizations of \(\lambda K_{h^u}\). We also establish the necessary and sufficient conditions for the existence of cube frames of \(\lambda K_{h^u}\).