On z-cycle factorizations with two associate classes where z is 2a and a is even

Let K = K ( a , p ; λ 1 , λ 2 ) be the multigraph with: the number of parts equal to p ; the number of vertices in each part equal to a ; 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, and when a ≡ 0 (mod z ) . In this paper, we give a construction for C z -factorizations of K for z = 2 a when a is even.