## Taiwanese Journal of Mathematics

### 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.

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 1 (2008), 137-149.

Dates
First available in Project Euclid: 21 July 2017

https://projecteuclid.org/euclid.twjm/1500602493

Digital Object Identifier
doi:10.11650/twjm/1500602493

Mathematical Reviews number (MathSciNet)
MR2387109

Zentralblatt MATH identifier
1177.05096

Keywords
cyclic cycle system

#### Citation

Wu, Shung-Liang; Lee, Dung-Ming. K-CYCLIC EVEN CYCLE SYSTEMS OF THE COMPLETE GRAPH. Taiwanese J. Math. 12 (2008), no. 1, 137--149. doi:10.11650/twjm/1500602493. https://projecteuclid.org/euclid.twjm/1500602493

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