\(4\)-Cycle Decompositions of the Cartesian Product of Two Complete Graphs

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 \(m, n \equiv 0 \pmod{2}\), \(m, n \equiv 1 \pmod{8}\), or \(m, n \equiv 5 \pmod{8}\).