New Group Divisible Designs With Block Size Four and Five

Abstract

Group divisible designs are classical combinatorial structures studied for their applications in construction of other combinatorial structures as well as a design of experiments. They provide an ample opportunity of developing techniques to study combinatorial structures and are inherently hard to construct, especially when the number of groups is less than the block size or when group sizes are different. The subject matter of the study is group divisible designs (GDDs) with block size 4 with 2 or 3 groups of different sizes, and with block size 5 with 2 groups of equal size. We present a general construction for GDDs with two groups of different sizes and block size k. Further, we present general constructions for such GDDs, where second associate pairs occur exactly once in the design and exactly twice in the design. We also present some interesting constructions of GDDs with block size 4 and three groups: one of size 1 and the other two of the same size n. We obtain necessary conditions for the existence of GDDs with 3 groups of sizes 1, n, n respectively and prove that the necessary conditions are sufficient for n is congruent to 0,1,4,5,8,9 (modulo 12) when the number of occurence of pairs of elements from the same group is greater or equal to the number of occurence of pairs of elements from different groups. Then we focus on two new problems of group divisible designs with block size 4. The first problem involves GDDs with three groups of sizes 1, n and n+2. We also obtain several non-existence results when restrictions on block configurations are placed. The second problem deals with group sizes 3, n and n+1 where n is greater or equal to 3. Finally, we study on the existence of 4-group divisible designs (4-GDDs) with two groups of equal size and blocks of size 5. We obtain necessary conditions for the existence of a 4-GDDs with 2 groups of size n and block size 5 and show that the necessary conditions are sufficient for group sizes 3,4,5,6,7,9 and 10. We apply combinatorial methods to obtain the necessary conditions and direct method to show the necessry conditions are sufficient.