C4-factorizations with two associate classes, λ1 is odd

Let K = K(a, p;λ1, λ2) 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 λ1; and the number of edges joining any two vertices of different parts equal to λ2. This graph was of interest to Bose and Shimamoto in their study of group divisible designs with two associate classes. Necessary and sufficient conditions for the existence of z-cycle decompositions of this graph have been found when z ∈ {3, 4}. The existence of C4-factorizations of K has been settled when a is even, and when a ≡ 1(mod 4) and λ1 is even. In this paper, necessary and sufficient conditions for the existence of a C4-factorization of K(a, p;λ1, λ2) are found when a ≡ 1(mod 4) and λ1 is odd with one possible exception.