A New Class Of Group Divisible Designs With Block Sizes From {3, 4~*}

Combinatorial design theory is an important branch of modern combinatorial the-ory．The research of design refers to a very important and central problem of combina-torial theory．Let K and G be sets of positive integers and letλbe a positive integer．A group divisible design of indexλand order v((K,λ)-GDD)is a triple(?),where V is a finite set of cardinality v,(?) is a partition of V into parts(groups)whosesizes lie in G,and (?) is a family of subsets(blocks)of V that satisfy:(1)if B∈(?) then |B|∈K,(2)|B∩G|≤1 for any B∈(?),G∈(?),(3)every pair of distinct elements of V occurs in exactlyλblocks or one group,butnot both．If K={k₁^*,k₂,…,k_n},we mean that there is at least one block of k₁ length in (?)．In this article,we investigated the existence problem of{3,4^*}-GDD of type g^t．This thesis is organized as follows．In Chapter 1,some basic conceptions and theorems are introduced．In Chapter 2,the existence of{3,4^*}-PBD is discussed．we prove that there ex-ists{3,4^*}-PBD if and only if v≡0,1(mod 3),v≥4 and v≠6,7,9．In Chapter 3,we focus on the existence of{3,4^*}-GDD of type g^t．There exists{3,4^*}-GDD of type g^t if g²t(t-1)≡0(mod 6),t≥4 and t=6 when g(?)1,5(mod 6)with the exception of(g,t)∈{(1,6),(1,7),(1,9),(2,4)}．In Chapter 4,the conclusion and some problems about t=6 and g≡1,5(mod 6)of{3,4^*}-GDD of type g^t are given．...