$4$-Cycle Decompositions of the Cartesian Product of Two Complete Graphs

Abstract

In this paper, we establish necessary and sufficient conditions on $m$ and $n$ in order for $K_m\times K_n$, the Cartesian product of two complete graphs, to be decomposable into cycles of length $4$. The main result is that $K_m\times K_n$ can be decomposed into cycles of length $4$ if and only if either $m,n\equiv 0(\mathrm{mod}2)$, $m,n\equiv 1(\mathrm{mod}8)$, or $m,n\equiv 5(\mathrm{mod}8)$.