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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
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[1] Baker, R.D.; Wilson, R.M., Nearly kirkman triple systems, Utilitas math., 11, 289-296, (1977) · Zbl 0362.05030
[2] Bose, R.C.; Parker, E.T.; Shrikhande, S., Further results on the construction of mutually orthogonal Latin squares and the falsity of the Euler’s conjecture, Can. J. math., 12, 189-203, (1960) · Zbl 0093.31905
[3] Bose, R.C.; Shrikhande, S., On the construction of sets of mutually orthogonal Latin squares and falsity of a conjecture of Euler, Trans. amer. math. soc., 95, 191-200, (1960) · Zbl 0093.31904
[4] Brower, A.E., Two nearly kirkman triple systems, Utilitas math., 13, 311-314, (1978) · Zbl 0379.05008
[5] Drake, D.A.; Larson, J., Pairwise balanced designs whose line sizes do not divide six, J. combin. theory, 34, A, 266-300, (1983) · Zbl 0518.05011
[6] Hanani, H., On resolvable balanced incomplete block designs, J. combin. theory, 17, A, 275-289, (1974) · Zbl 0305.05010
[7] Hanani, H., Balanced incomplete block designs and related designs, Discrete math., 11, 255-369, (1975) · Zbl 0361.62067
[8] Hanani, H.; Ray-Chaudhuri, D.K.; Wilson, R.M., On resolvable designs, Discrete math., 3, 75-97, (1972) · Zbl 0263.05016
[9] Kirkman, T.P., On a problem in combinations, Cambridge and Dublin math. J., 2, 191-204, (1847)
[10] Rees, R.; Stinson, D.R., On resolvable group divisible designs with block size 3, Ars combinatoria, 23, 107-120, (1987) · Zbl 0621.05004
[11] Ray-Chaudhuri, D.K.; Wilson, R.M., Solution of Kirkman’s schoolgirl problem, Combinatorics, Proc. sympos. pure math., 19, 187-203, (1971) · Zbl 0248.05009
[12] Stinson, D.R., Frames for kirkman triple systems, Discrete math., 62, 289-300, (1987) · Zbl 0651.05015
[13] Todorov, D.T., Three mutually orthogonal Latin squares of order 14, Ars combinatoria, 20, 45-47, (1985) · Zbl 0596.05009
[14] Wallis, W.D., Three orthogonal Latin squares, Congressus numerantium, 42, 69-86, (1984) · Zbl 0546.05011
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