On The Existence Of κ-cycle Frames With Uniform Group Size

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 A and order v, denoted by (J, A)-CGDD. The type of the CGDD(V, g,C) is the multiset of sizes|G| of the G ∈g and we usually use the "exponential" notation for its description:type 1i2i3k… denotes i occurrences of groups of size 1, j occurrences of groups of size 2, and so on.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 for some Gi∈g. Many authors contributed to the study of the existence of a ({k}, λ)-cycle frame of type gu(briefly, (k,λ)-CF(gu)). For k= 0 (mod 2), the existence of a (k, A)-cycle frame of type gu has been solved completely. For k= 1 (mod 2),3≤k≤13, the existence of a (k, λ)-cycle frame of type gu has been solved. For the left cases, by using recursive constructions, the necessary conditions for the existence of a (k,λ)-CF(gu) will be proved sufficient if there exist a (k,2)-CF(k5) and a (k,1)-CF(gu)for any(g,u)∈{(2,k+1),(2,2k+1),(2,3k+1),(4,k+1),(4,2k+1),(4,3k+ 1), (2k,5)}. M. Buratti, H. Cao and T. Traetta have completely settled most of the above 8 cases recently.In this thesis, we will settle the remaining cases and construct a(k, 1)-CF(gu) for any (g, u)∈{(2,3k+1), (4,2k+1), (4,3k+1)}. Thus, the existence of a (k,1)-CF(gu) will be solved completely.