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 $uh\equiv 0(\mathrm{mod}8)$, $\lambda (u-1)h\equiv 0(\mathrm{mod}3)$, and $u\ge 2$. 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 $\lambda K_{h^u}$.