Cyclic \(m\)-Cycle Systems of \(K_{n,n}\) for \(m \leq 30\)

Abstract

Let \(K_{n,n}\) denote the complete bipartite graph with \(n\) vertices in each part. In this paper, it is proved that there is no cyclic \(m\)-cycle system of \(K_{n,n}\) for \(m \equiv 2 \pmod{4}\) and \(n \equiv 2 \pmod{4}\). As a consequence, necessary and sufficient conditions are determined for the existence of cyclic \(m\)-cycle systems of \(K_{n,n}\) for all integers \(m \leq 30\).