$C_4$-Frames of $M(b,n)$

Abstract

Let $M(b,n)$ be the complete multipartite graph with $b$ parts $B_0,\dots ,B_{b-1}$ of size $n$. A $4$-cycle system of $M(b,n)$ is said to be a $frame$ if the $4$-cycles can be partitioned into sets $S_1,\dots ,S_z$ such that for $1\le j\le z$, $S_j$ induces a $2$-factor of $M(b,n)\setminus B_i$ for some $i\in {\mathbb{Z}}_b$. The existence of a $C_4$-frame of $M(b,n)$ has been settled when $n=4$ [6]. In this paper, we completely settle the existence question of a $C_4$-frame of $M(b,n)$ for all $b\ne 2$ and $n$.