Cycle Decompositions of $K_{\lowercase{n,n}}-I$

Abstract

Let $K_{n,n}$ denote the complete bipartite graph with $n$ vertices in each bipartition set and $K_{n,n}-I$ denote $K_{n,n}$ with a 1-factor removed. An $m$-cycle system of $K_{n,n}-I$ is a collection $T$ of $m$-cycles such that each edge of $K_{n,n}-I$ is contained in a unique $m$-cycle of $T$. In this paper, it is proved that the necessary and sufficient conditions for the existence of an $m$-cycle system of $K_{n,n}-I$ are $n\equiv 1 \({\rm 2)$, $m\equiv ({\rm 2)$, $4\leq m \leq 2n$, and $n(n-1)\equiv 0 ({\rm mod} m)$.