The classification of finite and locally finite connected-homogeneous digraphs

Abstract

We classify the finite connected-homogeneous digraphs, as well as the infinite locally finite such digraphs with precisely one end. This completes the classification of all the locally finite connected-homogeneous digraphs.

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References

  1. [1]

    P. J. Cameron, C. E. Praeger and N. C. Wormald: Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 13 (1993), 377–396.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    G. L. Cherlin: The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments, Mem. Amer. Math. Soc., no. 621, Amer. Math. Soc., 1998.

    Google Scholar 

  3. [3]

    R. Diestel, H. A. Jung and R. G. Möller: On vertex transitive graphs of infinite degree, Arch. Math. (Basel) 60 (1993), 591–600.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    H. Enomoto: Combinatorially homogeneous graphs, J. Combin. Theory (Series B) 30 (1981), 215–223.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    A. Gardiner: Homogeneous graphs, J. Combin. Theory (Series B) 20 (1976), 94–102.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    A. Gardiner: Homogeneity conditions in graphs, J. Combin. Theory (Series B) 24 (1978), 301–310.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    M. Goldstern, R. Grossberg and M. Kojman: Infinite homogeneous bipartite graphs with unequal sides, Discrete Math. 149 (1996), 69–82.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    Ja. Ju. Gol’fand and M. H. Klin: On k-homogeneous graphs, Algorithmic Stud. Combin. 186 (1978), 76–85 (in Russian).

    MathSciNet  MATH  Google Scholar 

  9. [9]

    R. Gray and D. Macpherson: Countable connected-homogeneous graphs, J. Combin. Theory (Series B) 100 (2010), 97–118.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    R. Gray and R. G. Möller: Locally-finite connected-homogeneous digraphs, Discrete Math. 311 (2011), 1497–1517.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    M. Hamann and F. Hundertmark: The classification of connected-homogeneous digraphs with more than one end, Trans. Am. Math. Soc. 365 (2013), 531–553.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    M. Hamann and J. Pott: Transitivity conditions in infinite graphs, Combinatorica 32 (2012), 649–688.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    S. Hedman and W. Y. Pong: Locally finite homogeneous graphs, Combinatorica 30 (2010), 419–434.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    B. Huppert: Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, vol. 134, Springer-Verlag, Berlin-New York, 1967.

    Google Scholar 

  15. [15]

    G. A. Jones: Triangular maps and non-congruence subgroups of the modular group, Bull. London Math. Soc. 11 (1979), 117–123.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    A. Kurosch: Die Untergruppen der freien Produkte von beliebigen Gruppen, Math. Ann. 109 (1934), 647–660.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    A. H. Lachlan: Finite homogeneous simple digraphs, Proceedings of the Herbrand symposium (Marseilles, 1981) (J. Stern, ed.), Stud. Logic Found. Math., vol. 107, North-Holland, 1982, 189–208.

    Google Scholar 

  18. [18]

    A. H. Lachlan: Countable homogeneous tournaments, Trans. Am. Math. Soc. 284 (1984), 431–461.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    A. H. Lachlan and R. Woodrow: Countable ultrahomogeneous undirected graphs, Trans. Am. Math. Soc. 262 (1980), 51–94.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    H. D. Macpherson: Infinite distance transitive graphs of finite valency, Combinatorica 2 (1982), 63–69.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    M. H. Millington: Subgroups of the classical modular group, J. London Math. Soc. 1 (1969), 351–357.

    MathSciNet  Article  MATH  Google Scholar 

  22. [22]

    J. Sheehan: Smoothly embeddable subgraphs, J. London Math. Soc. 9 (1974), 212–218.

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    W. W. Stothers: Subgroups of infinite index in the modular group, Glasgow Math. J. 19 (1978), 33–43.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    W. W. Stothers: Subgroups of infinite index in the modular group. II., Glasgow Math. J. 22 (1981), 101–118.

    MathSciNet  Article  MATH  Google Scholar 

  25. [25]

    W. W. Stothers: Subgroups of infinite index in the modular group. III., Glasgow Math. J. 22 (1981), 119–131.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Matthias Hamann.

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Hamann, M. The classification of finite and locally finite connected-homogeneous digraphs. Combinatorica 37, 183–222 (2017). https://doi.org/10.1007/s00493-015-2804-9

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Mathematics Subject Classification (2000)

  • 05C20
  • 05C25
  • 05C63