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A thousand million leagues

  • Tony Barnard (a1)

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Some time ago a friend, who was organising a chess league, rang me up and asked “How many different ways are there of arranging a roundrobin league tournament?” Thinking of his phone bill I told him I’d ring him back. Which was just as well, because the problem was much more difficult dian I had realised. Indeed it is unsolved in general. In the language of graph theory it is the problem of finding the number of onefactorisations of a complete graph. These and related objects have been the focus of considerable study both in combinatorics and recreational mathematics and, although the enumeration question remains unanswered in general, it is, even for small numbers of players, remarkably full of interest.

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1. Cameron, P.J., Parallelisms of complete designs, London Math. Soc. Lecture Note Ser. 23, C.U.P. (1976). Ch. 4: Edge-colourings of complete graphs.
2. Cameron, P.J., and Van Lint, J.H., Graphs, codes and designs, London Math. Soc. Lecture Note Ser. 43, C.U.P. (1980). Ch. 8: 1-factorisations of K6.
3. Coxeter, H.S.M., “Twelve points in PG(5,3) with 95040 self-transformations”. Proc. Roy. Soc. Ser. A, 247 (1958) 279293.
4. Dickson, L.E., and Safford, F.H., “Solution to Problem 8 (Group theory)Amer. Math. Monthly 13 (1906) 150151.
5. Gelling, E.N., On 1-factorizations of the complete graph and the relation to round robin schedules, MSc thesis, University of Victoria (1973).
6. Lucas, E., Récréations mathématiques. Vol. 2, Gauthier-Villars, Paris (1883). Sixiéme réécréation: Les jeux de demoiselles, pp161197.
7. Mendelsohn, E., and Rosa, A., “One factorizations of the complete graph - a survey”, J. Graph Theory 9 (1985) 4365.
8. Sylvester, J.J., Collected mathematical papers 1, C.U.P. (1904). No. 17: Elementary researches in the analysis of combinatorial aggregation (1844).
9. Sylvester, J.J., Collected mathematical papers 2, C.U.P. (1908). No. 46: Note on the historical origin of the unsymmetrical six-valued function of six letters (1861).
10. Wallis, W.D., Street, A.P., and Wallis, J.S., Combinatorics: Room squares, Sum-free sets, Hadamard matrices, Lecture Notes in Mathematics 292, Springer, Berlin (1972). Chapter VIII: Room squares of side 7.
11. Wallis, W.D., “One-factorizations of graphs: tournament applications”, College Math. Journal 18 (2) (1987) 116123.
12. Wallis, W.D., “A tournament problem”, J. Austral. Math. Soc. Ser. B 24 (1983) 289291.

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A thousand million leagues

  • Tony Barnard (a1)

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