On \(z\)-cycle factorizations with two associate classes where \(z\) is in \(\{4,4a\}\) with even parameters

Let \(K = K(a,p;λ_1,λ_2)\) 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 \(λ_1\); and the number of edges joining any two vertices of different parts equal to \(λ_2\). The existence of \(C_4\)-factorizations of \(K\) has been settled when \(a\) is even; when \(a ≡ 1 \ (mod 4)\) with one exception; and for very few cases when \(a ≡ 3 \ (mod 4)\). The existence of \(C_z\)-factorizations of \(K\) has been settled when \(a ≡ 1 \ (mod z)\) and \(λ_1\) is even; when \(a ≡ 0 \ (mod z)\); and when \(z=2a\) where both \(a\) and \(λ_1\) is even. In this paper, we give a construction for \(C_z\)-factorizations of \(K\) for \(z ∈ \{ 4,4a \}\) when \(a\) is even.