The Possible Number of Cycles in Cycle Systems

Abstract

For $v\ge 3$, $v$ odd, it is shown that there exists a decomposition of $K_v$ into $6$ cycles whose edges partition the edge set of $K_v$, if and only if

$$\lfloor \frac{v-1}{2}\rfloor \le b\lfloor \frac{v(v-1)}{6}\rfloor .$$
For even $v$, $v\ge 4$, a similar result is obtained for $K_v$ minus a $1$-factor.