4-cycle group-divisible designs with two associate classes (Q2757075)

The graph \((\lambda_1,\lambda_2) K_{(n,m)}\) has \(mn\) vertices; its vertex set is partitioned into \(m\) groups of size \(n\) each; two vertices are joined by \(\lambda_1\) or \(\lambda_2\) edges according to whether they both belong to the same group, or to different groups. In this paper, necessary and sufficient conditions are obtained for the existence of a decomposition of the graph \((\lambda_1,\lambda_2) K_{(n,m)}\) into 4-cycles for all integers \(m,n\geq 1\), and \(\lambda_1,\lambda_2\geq 0\).