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 G 1 , G 2 , … , G m . Let | V | = v and G = { G 1 , G 2 , … , G m } . If the edges of λ H 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 ∖ G i into cycles for some G i ∈ G . The existence of ( k , λ ) -cycle frames of type g u with 3 ≤ k ≤ 6 has been solved completely. In this paper, we show that there exists a ( { 3 , 5 } , λ ) -cycle frame of type g u for any u ≥ 4 , λ g ≡ 0 ( mod 2 ) , ( 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 ( 2 k t + 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 ( 2 k t + 1 ) for all t ≥ 1 , 11 ≤ k ≤ 49 , k ≡ 1 ( mod 2 ) and t ≠ 2 , 3 , 5 .