Abstract
A generalized balanced tournament design,GBTD(n, k) defined on akn-setV, is an arrangement of the blocks of a (kn, k, k−1)-BIBD defined onV into ann× (kn−1), array such that (1) every element ofV is contained in precisely one cell of each column, and (2) every element ofV is contained in at mostk cells of each row. In this paper, we completely determine the spectrum ofGBTD(n, 3). In addition we prove the exitence of factoredGBTD(n, 3) forn a positive integer,n≥4, with at most one possible exception.
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References
R.J.R. Abel,Four mutually orthogonal Latin squares of orders 28 and 52, J. Combinatorial Theory (A), to appear.
A. E. Brouwer,The number of mutually orthogonal Latin squares, Math. Centre report ZW 123, June 1979.
A.E. Brouwer and G.H.J. van Rees,More mutually orthogonal Latin squares, Discrete Mathematics 39 (1982) 263–281.
E.R. Lamken,A note on partitioned balanced tournament designs, Ars Combinatoria 24 (1987) 5–16.
E.R. Lamken,Generalized balanced tournament designs, Transactions of the American Mathematical Society 318, (1990) 473–490.
E.R. Lamken,Constructions for generalized balanced tournament designs, Discrete Mathematics, to appear.
E.R. Lamken,Constructions for resolvable and near resolvable (v, k, k−1)-BIBDs, Coding Theory and Design Theory, Part II, Design Theory, ed. D. Ray-Chaudhuri, Springer-Verlag, IMA Volume 21 (1990) 236–250.
E.R. Lamken,3-complementary frames and doubly near resolvable (v, 3, 2)-BIBDs, Discrete Mathematics 88 (1991) 59–78.
E.R. Lamken,The existence of 3 orthogonal partitioned incomplete Latin squares of type t n, Discrete Mathematics, 89 (1991) 231–251.
E.R. Lamken,On near generalized balanced tournament designs, Discrete Mathematics, 97 (1991) 279–294.
E.R. Lamken,A few more partitioned balanced tournament designs, preprint.
E.R. Lamken,The existence of doubly resolvable (v, 3, 2)-BIBDs, in preparation.
E.R. Lamken,The existence of partitioned generalized balanced tournament designs with block size 3, in preparation.
E.R. Lamken and S.A. Vanstone,The existence of factored balanced tournament designs, Ars Combinatoria 19(1985) 157–160.
E.R. Lamken and S.A. Vanstone,The existence of partitioned balanced tournament designs of side 4n +1, Ars Combinatoria 20 (1985) 29–44.
E.R. Lamken and S.A. Vanstone,The existence of partitioned balanced tournament designs of side 4n +3, Annals of Discrete Mathematics 34 (1987) 319–338.
E.R. Lamken and S.A. Vanstone,The existence of partitioned balanced tournament designs, Annals of Discrete Mathematics 34 (1987) 339–352.
E.R. Lamken and S.A. Vanstone,Balanced tournament designs and related topics, Discrete Mathematics 77 (1989) 159–176.
E.R. Lamken and S.A. Vanstone,Skew transversals in frames, J. of Combinatorial Math and Combinatorial Computing 2 (1987) 37–50.
E.R. Lamken and S.A. Vanstone,Existence results for doubly near resolvable (v, 3, 2)-BIBDs, Discrete Mathematics, to appear.
R. Mathon and A. Rosa,Tables of parameters of BIBDs with r ≤ 41 including existence, enumeration and resolvability results, Annals of Discrete Math 26 (1985) 275–308.
R.C. Mullin,Finite bases for some PBD-closed sets, Discrete Mathematics 77 (1989) 217–236.
E.T. Parker,Nonextendability conditions on mutually orthogonal Latin squares, Proceedings of the American Mathematical Society 13 (1962) 219–221.
R. Roth and M. Peters,Four pairwise orthogonal Latin squares of order 24, J. Combinatorial Theory (A) 44 (1987) 152–155.
P.J. Schellenberg, G.H.J. van Rees and S.A. Vanstone,The existence of balanced tournament designs, Ars Combinatoria 3 (1977) 303–318.
D.R. Stinson,The equivalence of certain incomplete transversal designs and frames, Ars Combinatoria 22 (1986) 81–87.
D.R. Stinson and L. Zhu,On sets of three MOLS with holes, Discrete Mathematics 54 (1985) 321–328.
D.T. Todorov,Three mutually orthogonal Latin squares of order 14 Ars Combinatoria 20 (1985) 45–48.
D.T. Todorov,Three mutually orthogonal Latin squares of order 20, Ars Combinatoria 27 (1989) 63–65.
S.A. Vanstone,Doubly resolvable designs, Discrete Mathematics 29 (1980) 77–86.
S.A. Vanstone,On mutually orthogonal resolutions and near resolutions, Annals of Discrete Mathematics 15 (1982) 357–369.
S.A. Vanstone,private communication.
R.M. Wilson,Concerning the number of mutually orthogonal Latin squares, Discrete Mathematics, (1974) 181–198.
L. Zhu,Pariwise orthogonal Latin squares with orthogonal small subsquares, Research report CORR 83-19, University of Waterloo, Waterloo, Ontario, Canada (1983).
L. Zhu,Incomplete transversal designs with block size five, Congressus Numerantium 69 (1989) 13–20.
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Communicated by S.A. Vanstone
This research was supported by a Postdoctoral membership at the Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455.
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Lamken, E.R. Existence results for generalized balanced tournament designs with block size 3. Des Codes Crypt 3, 33–61 (1993). https://doi.org/10.1007/BF01389354
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DOI: https://doi.org/10.1007/BF01389354