Existence of cyclic \(k\)-cycle systems of the complete graph (Q1861284)

The existence problems for cyclic \(k\)-cycle systems of the complete graph \(K_v\) with \(k \equiv 1\) (mod \(2k\)) and of the complete \(m\)-partite graph \(K_{m \times k}\) with odd \(m\) and \(k\) are considered. The authors give explicit constructions for all admissible cases using Rosa and Skolem sequences for suitable parameters. In particular, they prove that a cyclic \(p\)-cycle system of \(K_v\) with prime \(p\) exists for all admissible values of \(p\) and \(v\) but \((p,v)=(3,9)\).