Cycle Systems in the Complete Bipartite Graph Plus a One-Factor

Abstract

Let $K_{n,n}$ denote the complete bipartite graph with n vertices in each partite set and $K_{n,n}+I$ denote $K_{n,n}$ with a one-factor added. It is proved in this paper that there exists an m-cycle system of $K_{n,n}+I$ if and only if $n \equiv 1 (\rm{mod} 2)$, $m \equiv 0 (\rm{mod} 2)$, $4 \leq m \leq 2n$, and $n(n+1) \equiv$ 0 (mod m).