The Spectrum For Strictly Pairwise Distinct Group Divisible Designs

Let v and A be positive integers, let K and Mbe sets of positive integers. A pairwise distinct group divisible design is a triple (V,(?),B) such that:(1) V is a set of v points,(?) is a partition of a set V into subsets called group;(2) B is a set of subsets of V (called blocks);(3) each of cardinality from K, such that a group and a block contain at most one common point;(4)no two pair of points from distinct groups occur in the same number of blocks. Further, if for each1≤i≤v(v-1)/2, there exists exactly one pair of V occurring in i blocks, such pairwise distinct group divisible design is said to be strict.The necessary conditions for the existence of strictly pairwise distinct group divis-ible triple system are g=0(mod3), or g=1,2(mod3) and t=0,1(mod3). In this paper the existence of pairwise distinct group divisible design are investigated. It is established that the necessary conditions for the existence of a strictly pairwise distinct group divisible triple system are also sufficient.There are five chapters in this thesis:In Chapter1, we introduce the research background and some basic definitions of pairwise distinct group divisible design.In Chapter2, we introduce some useful lemmas, then we introduce three construc-tions which are very useful for the proof of the main conclusions in the thesis.In Chapter3, we present some examples of strictly pairwise distinct group divisible triple systems of small order, then we prove the necessary conditions for the existence of the strictly pairwise distinct group divisible triple system are also sufficient except forg=11,17,23,37.In Chapter4, we introduce some results of pairwise distinct group divisible design when the block size is four.In Chapter5, the main conclusions of this thesis are summarized, and the further research problems are presented at last.