The Existence Of(3,λ)-GDDs Of Type U~r1~t

Group divisible design (GDD) is one of the most basic and most important combinatorial structures. The necessary and sufficient conditions for the existence of a (3,1)-GDD of type urlt have been established by Colbourn et al. In this thesis, we investigate the existence of a (3, λ)-GDD of type urlt for any λ≥2. We prove that (3, λ)-GDDs of type urlt exist for the vast majority parameters (u, r, t, λ) which satisfy the necessary conditions through discussing the sufficience of some cases according to A=2,3,6. There are four chapters in this thesis.In chapter1, we introduce the background of (3, λ)-GDD, the definition and the known results. Some auxiliary designs are also introduced in this chapter.In chapter2, we mainly prove that the necessary conditions for the existence of a (3, λ)-GDD of type urlt and give some methods of direct constructions and recursive constructions. Moreover, we handle the existence of (3, λ)-GDD of small order utilizing some methods of construction.In chapter3, we mainly investigate the existence of a (3, λ)-GDD of type urlt and obtain the following results:(1) For λ=2, there is a (3,2)-GDD of type urlt whenever the necessary conditions are met;(2) For λ=3, there is a (3,3)-GDD of type urlt except the case r=5,7, when the necessary conditions are met;(3) For λ=6, the problem of the existence of (3,6)-GDD of type urlt is solved, except possibly13values, when the necessary condition are met;(4) Using the above results, we get a conclusion about the existence of (3, λ)-GDD of type urlt.In chapter4, the main results of this thesis are summarized, and finally the further research problems are presented.