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On Point-Symmetric Tournaments

  • Brian Alspach (a1)

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A tournament is a directed graph in which there is exactly one arc between any two distinct vertices. Let denote the automorphism group of T. A tournament T is said to be point-symmetric if acts transitively on the vertices of T. Let g(n) be the maximum value of taken over all tournaments of order n. In [3] Goldberg and Moon conjectured that with equality holding if and only if n is a power of 3. The case of point-symmetric tournaments is what prevented them from proving their conjecture. In [2] the conjecture was proved through the use of a lengthy combinatorial argument involving the structure of point-symmetric tournaments. The results in this paper are an outgrowth of an attempt to characterize point-symmetric tournaments so as to simplify the proof employed in [2].

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References

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1. Alspach, B., A class of tournaments, unpublished doctoral dissertation, University of California, Santa Barbara, 1966.
2. Alspach, B., A combinatorial proof of a conjecture of Goldberg and Moon, Canad. Math. Bull. 11 (1968), 655-661.
3. Goldberg, M. and Moon, J. W., On the maximum order of the group of a tournament, Canad. Math. Bull. 9 (1966), 563-569.
4. Moon, J. W., Tournaments with a given automorphism group, Canad. J. Math. 16 (1964) 485-489.
5. Rotman, J., The theory of groups: An introduction, Allyn and Bacon, Boston, 1966.
6. Turner, J., Point-symmetric graphs with a prime number of points, J. Comb. Theor. 3 (1967), 136-145.
7. Wielandt, H., Finite permutation groups, Trans. Bercov, R., Academic Press, New York, 1964.
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On Point-Symmetric Tournaments

  • Brian Alspach (a1)

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