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Resolvable group divisible designs with block size 3. (English) Zbl 0714.05007
Summary: Let $$\nu$$ be a non-negative integer, let $$\lambda$$ be a positive integer, and let K and M be sets of positive integers. A group divisible design, denoted by GD[K,$$\lambda$$,M,$$\nu$$ ], is a triple (X,$$\Gamma$$,$$\beta$$) where X is a set of points, $$\Gamma =\{G_ 1,G_ 2,...\}$$ is a partition of X, and $$\beta$$ is a class of subsets of X with the following properties. (Members of $$\Gamma$$ are called groups and members of $$\beta$$ are called blocks.)
1. The cardinality of X is $$\nu$$.
2. The cardinality of each group is a member of M.
3. The cardinality of each block is a member of K.
4. Every 2-subset $$\{$$ x,y$$\}$$ of X such that x and y belong to distinct groups is contained in precisely $$\lambda$$ blocks.
5. Every 2-subset $$\{$$ x,y$$\}$$ of X such that x and y belong to the same group is contained in no block.
A group divisible design is resolvable if there exists a partition $$\Pi =\{P_ 1,P_ 2,...\}$$ of $$\beta$$ such that each part $$P_ i$$ 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 $$\lambda$$. The case where $$M=\{1\}$$ has been solved by Ray-Chaudhuri and Wilson for $$\lambda =1$$, and by Hanani for all $$\lambda >1$$. The case where M is a singleton set, and $$\lambda =1$$ 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 $$\lambda >1$$. 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.

##### MSC:
 05B05 Combinatorial aspects of block designs
##### Keywords:
group divisible design; GD; resolvable
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##### References:
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