Group Divisible Packings And Coverings

Combinatorial design theory is an important branch of discrete mathematics, and it is devoted to the study of things according to specific requirements of arrangement and discusses the nature of a knowledge. Group divisible design is an important coneept and it is the foundation of many big question in the theory of Combinatorial Designs. During years of group divisible design’s research, many results have been obtained. In 1975, Hanani has solved a existence problem of {3}-GDD. For the given integers, if the corresponding designs is non-existence, we should construct some close relationship structures about it, i.e. group divisible packing (or covering).In 1968, Spencer examined all possible leaves among (3,1)-MGDP of type 1" to determine the packing number D1(3,1n). In 1991, Mendelsohn, Shalaby and Shen proved that a (3,λ)-MGDP of type 1" with any given leave exists. For general g, Yin determined the packing number Dλ(3,gn); however, he presented only one possible leave graph. In 1996, Billington and Lindner examined all possible leaves among (3,1)-MGDP of type gn. Therefore, in this paper, we should examine all possible leaves among (3,λ)-MGDP of type gn.In 1977, Bermond and Schonheim initiated the study of (K3+e,1)-GDD of type 1". In 1998, Hoffman and Kirkpatrick showed necessary and sufficient conditions for the existence of the (K3+e, λ)-GDD of type 1". In 2003, Chang, Lo Faro and Tripodi examined all possible leaves of a (K3+e,λ)-MGDP of type 1". Therefore, in this paper, we should examine all possible leaves among (K3+e,λ)-MGDP of type g".The study of covering numbers began with Fort and Hedlund in 1958, who deter-mined C1(3,1n). For general g, Heinrich and Yin determined C1(3,gn), and Yin and Wang determined Cλ(3,gn).However, they presented only one possible excess graph. Therefore, in this paper, we should examine all possible excesses among (3,λ)-MGDC of type g".In 2013, Chang, Lo Faro, Tripodi and Zhou examined all possible excesses of a (K3+e,λ)-MGDC of type 1". Therefore, in this paper, we should examine all possible excesses among (K3+e,λ)-MGDC of type g".There are four chapters in this thesis:In Chapter 1, we introduce some basic definitions of (AH, G)-packing (or covering), and present the research status and the existing results of (3,λ)-MGDP (or MGDC) or (K3+e,λ)-MGDP (or MGDC) of type g" and main research contents. In order to provide an important theoretical basis for the following proof, we consider related types of group divisible designs and adaptations of some standard constructions.In Chapter 2, we give the necessary conditions for the leave of (3,λ)-MGDP or (K3+e,λ)-MGDP of type g". For λ= 1, the problem of it has already been solved. When 2≤ λ≤ 7, we have examined all possible leaves among (3,λ)-MGDP of type g". For general A, we have completed the proof of it by constructions.In Chapter 3, we give the necessary conditions for the excess of (3,λ)-MGDC or (K3+e,λ)-MGDC of type gn. When 1≤ λ≤ 7, we have examined all possible excesses among (3,λ)-MGDC of type g". For general λ, we have completed the proof of it by constructions.In Chapter 4, the main conclusions of this thesis are summarized, and the further research problems are presented at last. Moreover, it is to be described in full detail in relation to the difficulty of problems to be solved.