Cycle systems in the complete bipartite graph minus a one-factor
In this paper the edge decomposition of \(K_{n,n}- I\) (the complete bipartite graph from which a 1-factor is removed) into a family of \(m\)-cycles is investigated. Necessary conditions for the existence of such a decomposition are that \(m\geq 4\) even, \(n\geq 3\) odd, \(m< 2n\) and \(m| n(n -1)\). The authors prove that these necessary conditions are sufficient if either \( m\equiv 2\text{\,mod\,}4\) and \(m\leq 2n\) or \(m\equiv 0\text{\,mod\,}4\) and \(m\leq n\) (leaving open the case when \(m\equiv 0\text{\,mod\,} 4\) and \(n< m< 2n\)). The proof cleverly uses the concept of Cayley graphs and difference constructions.
scientific article; zbMATH DE number 2093765