Locally finite homogeneous graphs


A connected graph G is said to be z-homogeneous if any isomorphism between finite connected induced subgraphs of G extends to an automorphism of G. Finite z-homogeneous graphs were classified in [17]. We show that z-homogeneity is equivalent to finite-transitivity on the class of infinite locally finite graphs. Moreover, we classify the graphs satisfying these properties. Our study of bipartite z-homogeneous graphs leads to a new characterization for hypercubes.

This is a preview of subscription content, access via your institution.


  1. [1]

    M. R. Alfuraidan and J. I. Hall: Smith’s Theorem and a characterization of the 6-cube as distance-transitive graph, Journal of Algebraic Combinatorics24(2) (2006), 195–207.

    Article  MathSciNet  Google Scholar 

  2. [2]

    A. S. Asratian, T. M. J. Denley and R. Häggkvist: Bipartite Graphs and their Applications, Cambridge University Press, 1998.

  3. [3]

    H. J. Bandelt and V. Chepoi: Metric graph theory and geometry: a survey; Contemporary Mathematics453 (2008), 49–86.

    MathSciNet  Google Scholar 

  4. [4]

    A. E. Brouwer, A. M. Cohen and A. Neumaier: Distance Regular Graphs, Berlin, New York, Springer-Verlag, 1989.

    Google Scholar 

  5. [5]

    P. J. Cameron: 6-transitive graphs, J. Comb. Theory, Ser. B28 (1980), 168–179.

    Article  MATH  Google Scholar 

  6. [6]

    P. J. Cameron: Automorphism groups of graphs, in: Selected Topics in Graph Theory II, pp. 89–127, Academic Press, 1983.

  7. [7]

    H. Enomoto: Combinatorially homogeneous graphs, J. Comb. Theory, Ser. B30(2) (1981), 215–223.

    Article  MathSciNet  MATH  Google Scholar 

  8. [8]

    A. Gardiner: Homogeneous graphs, J. Comb. Theory, Ser. B20 (1976), 94–102.

    Article  MathSciNet  MATH  Google Scholar 

  9. [9]

    A. Gardiner: Homogeneous graphs stability, J. Austral. Math. Soc., Ser. A21(3) (1976), 371–375.

    Article  MathSciNet  MATH  Google Scholar 

  10. [10]

    A. Gardiner: Homogeneity conditions in graphs, J. Comb. Theory, Ser. B24(3) (1978), 301–310.

    Article  MathSciNet  MATH  Google Scholar 

  11. [11]

    A. Gardiner and C. E. Praeger: Distance-Transitive graphs of valency five, Proceedings of the Edinburgh Mathematical Society30 (1987), 73–81.

    Article  MathSciNet  MATH  Google Scholar 

  12. [12]

    C. Godsil and G. Royle: Algebraic Graph Theory, Graduate Texts in Mathematics, 207. Springer-Verlag, New York, 2001.

    Google Scholar 

  13. [13]

    R. Gray and H. D. MacPherson: Countable connected-homogeneous graphs, J. Comb. Theory, Ser. B100(2) (2010), 97–118.

    Article  MathSciNet  MATH  Google Scholar 

  14. [14]

    S. Hedman and W. Y. Pong: Quantifier-eliminable locally finite graphs, Mathematical Logic Quarterly, submitted.

  15. [15]

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

    MathSciNet  MATH  Google Scholar 

  16. [16]

    H. D. MacPherson: Infinite distance-transitive graphs of finite valency, Combinatorica2(1) (1982), 63–70.

    Article  MathSciNet  MATH  Google Scholar 

  17. [17]

    R. Weiss: Glatt einbettbare Untergraphen, J. Comb. Theory, Ser. B21 (1976), 276–281.

    Article  Google Scholar 

  18. [18]

    H. P. Yap: Some Topics in Graph Theory, Cambridge University Press, 1986.

Download references

Author information



Corresponding author

Correspondence to Shawn Hedman.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hedman, S., Pong, W.Y. Locally finite homogeneous graphs. Combinatorica 30, 419–434 (2010). https://doi.org/10.1007/s00493-010-2472-8

Download citation

Mathematics Subject Classification (2000)

  • 05C75