Resolvable group divisible designs with block size 3.

*(English)*Zbl 0714.05007Summary: 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.

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 |

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\textit{A. M. Assaf} and \textit{A. Hartman}, Discrete Math. 77, No. 1--3, 5--20 (1989; Zbl 0714.05007)

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##### References:

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