zbMATH — the first resource for mathematics

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.

05B05 Combinatorial aspects of block designs
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.