C4-Factorizations with Two Associate Classes

Abstract

Let $K = K(a,p;\lambda_{1},\lambda_{2})$ be the multigraph with: the number of vertices in each part equal to $a$; the number of parts equal to $p$; the number of edges joining any two vertices of the same part equal to $\lambda_{1}$; and the number of edges joining any two vertices of different parts equal to $\lambda_{2}$. This graph was of interest to Bose and Shimamoto in their study of group divisible designs with two associate classes \cite{bose}. Necessary and sufficient conditions for the existence of $z$-cycle decompositions of this graph have been found when $z \in \{3,4\}$\cite{fu1,fu2}. The existence of resolvable 4-cycle decompositions of $K$ has been settled when $a$ is even \cite{br}, but the odd case is much more difficult. In this paper, necessary and sufficient conditions for the existence of a $C_{4}$-factorization of $K(a,p;\lambda_{1},\lambda_{2})$ are found when $a \equiv 1 (mod\;4)$ and $\lambda_{1}$ is even, and all cases with one exception have been solved when $\lambda_{1}$ is odd.