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The Asymmetry Number of Finite Tournaments, and Some Related Results

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Abstract

We introduce the concept of asymmetry number for finite tournaments, as a natural generalization of that for graphs by Erdős and Rényi (Acta Math Acad Sci Hungar 14:295–315, 1963). We prove an upper bound for the asymmetry number of finite tournaments and discuss its asymptotically best possibility. We also give an alternative proof of a theorem by Jaligot and Khelif (Aequat Math 67:73–79, 2004) which states that the random tournament can be expressed by a Cayley tournament on an arbitrary countable group without involutions. We slightly improve a result by Cameron and Johnson (Math Proc Camb Philos Soc 102:223–231, 1987).

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References

  1. Bollobás, B.: Random Graphs, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 73. Cambridge University Press, Cambridge (2001)

  2. Cameron, P.J.: The random graph revisited. In: European Congress of Mathematics, vol. I (Barcelona, 2000) Progr. Math., vol. 201, pp. 267–274 (2000)

  3. Cameron, P.J., Johnson, K.W.: An investigation of countable B-groups. Math. Proc. Camb. Philos. Soc. 102, 223–231 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Diestel, R: Graph Theory, 4th edn. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2010)

  5. Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Acad. Sci. Hungar. 14, 295–315 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  6. Erdős, P., Spencer, J.: Probabilistic Methods in Combinatorics. Probability and Mathematical Statistics, vol. 17. Academic Press, New York (1974)

  7. Jaligot, E., Khelif, A.: The random tournament as a Cayley tournament. Aequat. Math. 67, 73–79 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kim, J.H., Sudakov, B., Vu, V.H.: On the asymmetry of random regular graphs and random graphs. Random Struct. Algorithm 21, 216–224 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. McDiarmid, C.: Concentration. In: Probabilistic methods for algorithmic discrete mathematics. Algorithms Combin., vol. 16, Springer, Berlin, pp. 195–248 (1998)

  10. McKay, B.: Combinatorial data. http://users.cecs.anu.edu.au/~bdm/data/. Accessed 28 Oct 2016

  11. Reid, K.B., Brown, E.: Doubly regular tournaments are equivalent to skew Hadamard matrices. J. Combin. Theory Ser. A 12, 332–338 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. Satake, S., Sawa, M., Jimbo, M.: Erdős–Rényi theory for asymmetric digraphs (in preparation)

  13. Spencer, J.: Maximal asymmetry of graphs. Acta Math. Acad. Sci. Hungar. 27, 47–53 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. Van Lint, J.H., Wilson, R.M.: A Course in Combinatorics, 2nd edn. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  15. Wright, E.M.: Asymmetric and symmetric graphs. Glasgow Math. J. 15, 69–73 (1963)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Masanori Sawa for his careful reading of this paper and many valuable comments and suggestions.

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  1. Graduate School of Information Science, Nagoya University Furo, Chikusa, Nagoya, 464-8601, Japan

    Shohei Satake

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  1. Shohei Satake
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Correspondence to Shohei Satake.

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Satake, S. The Asymmetry Number of Finite Tournaments, and Some Related Results. Graphs and Combinatorics 33, 1433–1442 (2017). https://doi.org/10.1007/s00373-017-1787-2

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  • DOI: https://doi.org/10.1007/s00373-017-1787-2

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