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

Let λ1Kaλ1Ka denote the graph on a   vertices with λ1λ1 edges between every pair of vertices. Take p   copies of this graph λ1Kaλ1Ka, and join each pair of vertices in different copies with λ2λ2 edges. The resulting graph is denoted by K(a,p;λ1,λ2)K(a,p;λ1,λ2), 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∈{3,4}z∈{3,4}. In this paper we consider resolvable decompositions, finding necessary and sufficient conditions for a 4-cycle factorization of K(a,p;λ1,λ2)K(a,p;λ1,λ2) (when λ1λ1 is even) or of K(a,p;λ1,λ2)K(a,p;λ1,λ2) minus a 1-factor (when λ1λ1 is odd) whenever a is even.