Some Results On Super-simple Cycle Group Divisible Designs

Let K be a set of positive integers. Suppose m>1and 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 K, then (V, G, C) is called a cycle group divisible design with index λ and order v, denoted by (K, λ)-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:type1i2j3k···denotes i occurrences of groups of size1, j occurrences of groups of size2, and so on.A (K, λ)-CGDD is said to be super-simple if any two distinct cycles C1, C2∈C have at most two points in common. Gronau and Mullin introduced the concept of super-simple designs in1992. Super-simple designs can be used to construct superimposed codes and other combinatorial designs.The existence of a (4,2)-SCGDD(1u) has been solved by Billington et al. re-cently. In this thesis, we shall prove that the necessary conditions for the existence of a (4, λ)-SCGDD(gu) for λ=2,3and a (4,4)-SCGDD(1u) are also sufficient.