The Possible Number of Cycles in Cycle Systems

For \(v \geq 3\), \(v\) 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

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