On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m -partite graph K m ( n 1 , n 2 , … , n m ) with vertex set V and m independent sets G 1 , G 2 , … , G m of n 1 , n 2 , … , n m vertices respectively. Let G = { G 1 , G 2 , … , G m } . If the edges of λ H can be partitioned into a set C of k -cycles, then ( V , G , C ) is called a k -cycle group divisible design with index λ , denoted by ( k , λ ) -CGDD. A ( k , λ ) -cycle frame is a ( k , 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 for some G i ∈ G . Stinson et al. have resolved the existence of ( 3 , λ ) -cycle frames of type g u . In this paper, we show that there exists a ( k , λ ) -cycle frame of type g u for k ∈ { 4 , 5 , 6 } if and only if g ( u − 1 ) ≡ 0 ( mod k ) , λ g ≡ 0 ( mod 2 ) , u ≥ 3 when k ∈ { 4 , 6 } , u ≥ 4 when k = 5 , and ( k , λ , g , u ) ≠ ( 6 , 1 , 6 , 3 ) . 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 ) . Lindner et al. have considered the general existence problem of k -ARCS ( n ) from the commutative quasigroup for k ≡ 0 ( mod 2 ) . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k -ARCS ( n ) s when k ≡ 1 ( mod 2 ) . We also update the known results and prove 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 known exceptions and four additional possible exceptions.