Resolvable group divisible designs with block size 3

Abstract

Let v be a non negative integer, let λ be a positive integer, and let K and M be sets of positive integers. A group divisible design, denoted by GD[K, λ, M, v], is a triple (X, G{cyrillic}, β) where X is a set of points, G{cyrillic} = {G1, G2,...} is a partition of X, and β is a class of subsets of X with the following properties. (Members of G{cyrillic} are called groups and members of β are called blocks.) 1. 1. The cardinality of X is v. 2. 2. The cardinality of each group is a member of M. 3. 3. The cardinality of each block is a member of K. 4. 4. Every 2-subset {x, y} of X such that x and y belong to distinct groups is contained in 5. precisely λ blocks. 6. 5. Every 2-subset {x, y} of X such that x and y belong to the same group is contained in no 7. block. A group divisible design is resolvable if there exists a partition Π = {P1, P2,...} of β such that each part Pi is itself a partition of X. In this paper we investigate the existence of resolvable group divisible designs with K = {3}, M a singleton set, and all λ. The case where M = {1} has been solved by Ray-Chaudhuri and Wilson for λ = 1, and by Hanani for all λ > 1. The case where M is a singleton set, and λ = 1 has recently been investigated by Rees and Stinson. We give some small improvements to Rees and Stinson's results, and give new results for the cases where λ > 1. We also investigate a class of designs, introduced by Hanani, which we call frame resolvable group divisible designs and prove necessary and sufficient conditions for their existence. © 1989.