Resolvable 4-cycle group divisible designs with two associate classes: Part size even

Let ${\lambda }_1{K}_a$ denote the graph on a vertices with ${\lambda }_1$ edges between every pair of vertices. Take p copies of this graph ${\lambda }_1{K}_a$, and join each pair of vertices in different copies with ${\lambda }_2$ edges. The resulting graph is denoted by $K(a,p;{\lambda }_1,{\lambda }_2)$, a graph that was of particular interest to Bose and Shimamoto in their study of group divisible designs with two associate classes. The existence of z-cycle decompositions of this graph have been found when $z\in \{3,4\}$. In this paper we consider resolvable decompositions, finding necessary and sufficient conditions for a 4-cycle factorization of $K(a,p;{\lambda }_1,{\lambda }_2)$ (when ${\lambda }_1$ is even) or of $K(a,p;{\lambda }_1,{\lambda }_2)$ minus a 1-factor (when ${\lambda }_1$ is odd) whenever a is even.