K-CYCLIC EVEN CYCLE SYSTEMS OF THE COMPLETE GRAPH

Abstract

An $(m_{1},\ldots,m_{r})$-cycle is the union of edge-disjoint $m_{i}$-cycles for $1\le i\le r$. An $(m_{1},\ldots,m_{r})$-cycle system of the complete graph $K_{v}$, $(\pmb{V},\pmb{C})$, is said to be $k$-cyclic if $\pmb{V}=Z_{v}$ and for $k\in Z_{v}$, $C+k\in \pmb{C}$ whenever $C\in \pmb{C}$. Let $m_{i}$ ($1\le i\le r$ ) be even integers ($\gt 2$) and let $\sum_{i=1}^{r}m_{i}=m=ks$ with $\gcd(k,s)=1$ and $k$ odd. Suppose $v$ is the least positive integer such that $v(v-1)\equiv0\pmod{2m}$ and $\gcd(v,m)=k.$ In this paper, it is proved that if there is a $k$-cyclic $(m_{1},\ldots,m_{r})$-cycle system of order $v$, then for any positive integer $p$, a $k$-cyclic $(m_{1},\ldots,m_{r})$ cycle system of order $2pm+v$ exists. As the main consequence of this paper, the necessary and sufficient conditions for the existence of a $k$-cyclic $(m_{1},\ldots,m_{r})$-cycle system of order $v$ with $m_{i}$ even and $\sum_{i=1}^{r}m_{i}\le20$ are given.