On Cycle Frames with Cycles of Length $8$

Abstract

Let $M(b,n)$ be the complete multipartite graph with $b$ parts $B_0,\dots ,B_{b-1}$ of size $n$. A $z$-cycle system of $M(b,n)$ is said to be a $cycle-frame$ if the $z$-cycles can be partitioned into sets $S_1,\dots ,S_k$ such that for $1\le j\le k$, $S_j$ induces a $2$-factor of $M(b,n)\mathrm{\setminus }{B}_i$ for some $i\in {\mathbb{Z}}_b$. The existence of a $C_z$-frame of $M(b,n)$ has been settled when $z\in \{3,4,5,6\}$. Here, we completely settle the case of $C_z$-frames when $z$ is $8$, and we give some solutions for larger values of $z$.