Combinatorial Properties of Group Divisible Incomplete Block Designs

Abstract

Group divisible incomplete block designs are an important subclass of partially balanced designs [1], [2] with two associate classes, and they may also be regarded as a special case of intra- and inter-group balanced incomplete block designs [3], [4]. They may be defined as follows. An incomplete block design with $v$ treatments each replicated $r$ times in $b$ blocks of size $k$ is said to be group divisible (GD) if the treatments can be divided into $m$ groups, each with $n$ treatments, so that the treatments belonging to the same group occur together in $\lambda_1$ blocks and treatments belonging to different groups occur together in $\lambda_2$ blocks. If $\lambda_1 = \lambda_2 = \lambda$ (say), then every pair of treatments occurs in $\lambda$ blocks, and the design becomes a balanced incomplete block design, which has been extensively studied [5], [6], [7], [8]. We shall therefore confine ourselves to the case $\lambda_1 \neq \lambda_2$. The object of this paper is to study the combinatorial properties of these designs. It is shown that the GD designs can be divided into three exhaustive and mutually exclusive classes: (a) Singular GD designs characterized by $r - \lambda_1 = 0$; (b) Semi-regular GD designs characterized by $r - \lambda_1 > 0, rk - v\lambda_2 = 0$; (c) Regular GD designs characterized by $r - \lambda_1 > 0, rk - v\lambda_2 > 0$. Certain inequality relations between the parameters necessary for the existence of the design have been derived in each case. Some other interesting theorems about the structure of these designs have also been obtained. Methods of constructing GD designs will be given in a separate paper.