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A Note on Min–Max Pair in Tournaments

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Abstract

We prove the following results in this paper. Let T be a tournament of order \(n\ge 3\) with vertex set V and arc set E, x a vertex of maximum out-degree and y a vertex of minimum out-degree of T. If \(yx\in E\) then there exists a path of length i from x to y for any i with \(2\le i \le n-1\); and if \(xy\in E\), then there exists a path of length i from x to y for any i with \(3\le i \le n-1\) unless xy is exceptional. We also give a very short discussion to almost regular tournaments and prove a result of Jakobsen.

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References

  1. Alspach, B.: Cycles of each length in regular tournaments. Can. Math. Bull. 10, 283–285 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alspach, B., Reid, K.B., Roselle, D.P.: Bypasses in asymmetric digraphs. J. Comb. Theory Ser. B 17, 11–18 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bang-Jensen, J., Gutin, G.: Digraphs Theory, Algorithms and Applications. Springer, Berlin (2007)

    MATH  Google Scholar 

  4. Havet, F.: Pancyclic arcs and connectivity in tournaments. J. Graph Theory 47(2), 87–110 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jakobsen, O.S.: Cycles and paths in tournaments. Thesis, University of Aarhus (1972)

  6. Lu, X.: On min–max pair in tournaments. Graphs Comb. 34, 613–618 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Thomassen, C.: hamiltonian-connected tournaments. J. Comb. Theory Ser. B 28, 142–163 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Tian, F., Zhang, C.-Q.: A note about the paths of each length in tournaments. J. Qufu Teach. Coll. 1, 19–21 (1981)

    Google Scholar 

  9. Yeo, A.: The number of pancyclic arcs in a k-strong tournament. J. Graph Theory 50, 212–219 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is very grateful to Dr. Xiaolan Hu for her careful reading and many excellent suggestions, and to referees for improvement.

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  1. Center for Statistical Research and Methodology, Census Bureau, Suitland, MD, USA

    Xiaoyun Lu

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  1. Xiaoyun Lu
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Correspondence to Xiaoyun Lu.

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Lu, X. A Note on Min–Max Pair in Tournaments. Graphs and Combinatorics 35, 1139–1145 (2019). https://doi.org/10.1007/s00373-019-02062-x

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  • DOI: https://doi.org/10.1007/s00373-019-02062-x

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