Some constructions for cyclic perfect cycle systems

We present direct and recursive constructions for some classes of regular (mostly cyclic) $i$-perfect $(\Gamma,C_k)$-designs with $\Gamma=K_v$ and $v\equiv1$ (mod $2k$) or $\Gamma=K_{m\times k}$ (the {\it complete $m$-partite graph with parts of size $k$}) and $m$ odd. In particular, we will get some classes of cyclic Steiner $k$-cycle systems and cyclic Kirkman $k$-cycle systems. We also prove the existence of a 2-perfect (though not regular) $k$-cycle system of order $v=km$ for any pair $(k,m)$ with $k\in\{15, 21,25,27,33,35,39\}$ and $m\equiv\pm1$ (mod $6$) with the only possible exception of $(k,m)=(27,5)$.