Even \((m_1, m_2,…,m_r)\)-Cycle Systems of the Complete Graph

Abstract

Let \(K_n\) be the complete graph on \(n\) vertices. In this paper, we find the necessary and sufficient conditions for the existence of an \((m_1, m_2, \ldots, m_r)\)-cycle system of \(K_n\), where \(m_i\) (\(1 \leq i \leq r\)) are positive even integers, and \(\sum_{i=1}^{r}m_i = 2^k\) for \(k \geq 2\). In particular, if \(r = 1\) then there exists a cyclic \(2^k\)-cycle system of \(K_n\) if and only if \(2^k\) divides \(|E(K_n)|\) and \(n\) is odd.