Whist tournaments with the three person property. (English) Zbl 1043.05016
Summary: A whist tournament Wh\((v)\) is said to have the three person property if any two games in the tournament do not have three common players. We denote such a design as 3PWh\((v)\). It is proved in the literature that a 3PWh\((v)\) exists for any \(v \geqslant 8\) and \(v\equiv 0,1 \pmod 4\) with 38 possible exceptions. In this paper, we resolve these open cases and complete the spectrum problem on the existence of 3PWh\((v)\)s.
MSC:
| 05B05 | Combinatorial aspects of block designs |
Keywords:
Whist tournamentReferences:
| [1] | Abel, R. J.R.; Bennett, F. E.; Zhang, H.; Zhu, L., Existence of HSOLSSOMs with type \(h^n\) and \(1^nu^1\), Ars. Combin., 55, 97-115 (2000) · Zbl 0993.05031 |
| [2] | Abel, R. J.R.; Brouwer, A. E.; Colbourn, C. J.; Dinitz, J. H., Mutually orthogonal Latin squares, (Colbourn, C. J.; Dinitz, J. H., C.R.C. Handbook of Combinatorial Designs (1996), CRC Press: CRC Press Boca Raton, FL), 111-142 · Zbl 0849.05009 |
| [3] | Anderson, I., Combinatorial Designs and Tournaments (1997), Oxford University Press: Oxford University Press Oxford · Zbl 0881.05001 |
| [4] | Anderson, I.; Finizio, N. J., Triplewhist tournaments that are also Mendelsohn designs, J. Combin. Designs, 5, 397-406 (1997) · Zbl 0914.05007 |
| [5] | R.D. Baker, Factorization of graphs, Doctoral Thesis, Ohio State University, 1975.; R.D. Baker, Factorization of graphs, Doctoral Thesis, Ohio State University, 1975. |
| [6] | Beth, T.; Jungnickel, D.; Lenz, H., Design Theory (1999), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0945.05005 |
| [7] | Bennett, F. E.; Zhang, H.; Zhu, L., Holey self-orthogonal Latin squares with symmetric orthogonal mates, Ann. Combin., 1, 107-118 (1997) · Zbl 0929.05012 |
| [8] | Bennett, F. E.; Zhu, L., Further results on the existence of HSOLSSOM \((h^n)\), Austral. J. Combin., 14, 207-220 (1996) · Zbl 0869.05015 |
| [9] | Bennett, F. E.; Zhu, L., The spectrum of \(HSOLSSOM (h^n )\) where \(h\) is even, Discrete Math., 158, 11-25 (1996) · Zbl 0867.05011 |
| [10] | Chen, K., On the existence of super-simple \((v,4,3)\)-BIBDs, J. Combin. Math. Combin. Comput., 17, 149-159 (1995) · Zbl 0827.05009 |
| [11] | Chen, K., On the existence of super-simple \((u,4,4)\)-BIBDs, J. Statist. Plann. Inference, 51, 339-350 (1996) · Zbl 0849.05007 |
| [12] | Finizio, N. J., Whist tournaments-three person property, Discrete Appl. Math., 45, 125-137 (1993) · Zbl 0793.90111 |
| [13] | Finizio, N. J., A few more Z-cyclic whist tournaments, J. Combin. Math. Combin. Comput., 19, 93-95 (1995) · Zbl 0839.05016 |
| [14] | Finizio, N. J.; Lewis, J. T., A criterion for cyclic whist tournaments with the three person property, Utilitas Math., 52, 129-140 (1997) · Zbl 0903.05010 |
| [15] | Furino, S. C.; Miao, Y.; Yin, J., Frames and Resolvable Designs (1996), CRC Press: CRC Press Boca Raton · Zbl 0855.62061 |
| [16] | Ge, G.; Abel, R. J.R., Some new HSOLSSOMs of types \(h^n\) and \(1^mu^1\), J. Combin. Designs, 9, 435-444 (2001) · Zbl 0994.05043 |
| [17] | G. Ge, C.W.H. Lam, Super-simple resolvable balanced incomplete block designs with block size 4 and index 3, J. Combin. Designs, to appear.; G. Ge, C.W.H. Lam, Super-simple resolvable balanced incomplete block designs with block size 4 and index 3, J. Combin. Designs, to appear. · Zbl 1031.05023 |
| [18] | Gronau, H.-D. O.F.; Mullin, R. C., On super-simple \(2-(v,4, λ)\) designs, J. Combin. Math. Combin. Comput., 11, 113-121 (1992) · Zbl 0769.05018 |
| [19] | A. Hartman, Doubly and orthogonally resolvable quadruple systems, Combinatorial mathematics, Proceedings of the Seventh Australian Conference, University of Newcastle, Vol. VII, Newcastle, 1979, pp. 157-164, Lecture Notes in Mathematics, Vol. 829, Springer, Berlin, 1980.; A. Hartman, Doubly and orthogonally resolvable quadruple systems, Combinatorial mathematics, Proceedings of the Seventh Australian Conference, University of Newcastle, Vol. VII, Newcastle, 1979, pp. 157-164, Lecture Notes in Mathematics, Vol. 829, Springer, Berlin, 1980. · Zbl 0466.05025 |
| [20] | Lu, Y.; Zhang, S. Y., Existence of whist tournaments with the three-person property \(3 PWh (v)\), Discrete Appl. Math., 101, 207-219 (2000) · Zbl 0944.05011 |
| [21] | Moore, E. H., Tactical Memoranda I-III, Amer. J. Math., 18, 264-303 (1896) · JFM 27.0472.02 |
| [22] | Mullin, R. C.; Stinson, D. R., Holey SOLSSOMs, Utilitas Math., 25, 159-169 (1984) · Zbl 0553.05021 |
| [23] | Mullin, R. C.; Zhu, L., The spectrum of HSOLSSOM \((h^n)\) where \(h\) is odd, Utilitas Math., 27, 157-168 (1985) · Zbl 0597.05014 |
| [24] | Wilson, R. M., Constructions and uses of pairwise balanced designs, Math. Centre Tracts, 55, 18-41 (1995) · Zbl 0312.05010 |
| [25] | Zhu, L., Existence for holey SOLSSOM of type \(2^n\), Congr. Numer., 45, 295-304 (1984) · Zbl 0589.05030 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
