Cyclic $m$-Cycle Systems of $K_{n,n}$ for $m\le 30$

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 $m\equiv 2(\mathrm{mod}4)$ and $n\equiv 2(\mathrm{mod}4)$. As a consequence, necessary and sufficient conditions are determined for the existence of cyclic $m$-cycle systems of $K_{n,n}$ for all integers $m\le 30$.