More results on cycle frames and almost resolvable cycle systems

Let J be a set of positive integers. Suppose m>1 and H is a complete m-partite graph with vertex set V and m groups G1,G2,…,Gm. Let |V|=v and G={G1,G2,…,Gm}. If the edges of λH can be partitioned into a set C of cycles with lengths from J, then (V,G,C) is called a cycle group divisible design with index λ and order v, denoted by (J,λ)-CGDD. A (J,λ)-cycle frame is a (J,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V∖Gi into cycles for some Gi∈G. The existence of (k,λ)-cycle frames of type gu with 3≤k≤6 has been solved completely. In this paper, we show that there exists a ({3,5},λ)-cycle frame of type gu for any u≥4, λg≡0(mod2), (g,u)≠(1,5),(1,8) and (g,u,λ)≠(2,5,1). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). It has been proved that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three exceptions and four possible exceptions. In this paper, we shall show that there exists a k-ARCS(2kt+1) for all t≥1, 11≤k≤49, k≡1(mod2) and t≠2,3,5.