Distinguishing graphs with intermediate growth


A graph G is said to be 2-distinguishable if there is a 2-labeling of its vertices which is not preserved by any nontrivial automorphism of G. We show that every locally finte graph with infinite nite motion and growth at most \(\mathcal{O}\left( {2^{(1 - \varepsilon )\tfrac{{\sqrt n }} {2}} } \right)\) is 2-distinguishable. Infinite motion means that every automorphism moves infinitely many vertices and growth refers to the cardinality of balls of radius n.

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


  1. [1]

    M. O. Albertson and K. L. Collins: Symmetry breaking in graphs, Electron. J. Combin. 3, Research Paper 18, 1996.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    M. Conder and T. Tucker: Motion and distinguishing number two, Ars Math. Contemp. 4 (2011), 63–72.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    J. Cuno, W. Imrich and F. Lehner: Distinguishing graphs with infinite motion and nonlinear growth. Ars Math. Contemp. 7 (2014), 201–213.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    R. Diestel: Graph theory, volume 173 of Graduate Texts in Mathematics, Springer- Verlag, Berlin, third edition, 2005.

    Google Scholar 

  5. [5]

    M. J. Fisher and G. Isaak: Distinguishing numbers of Cartesian products of mul-tiple complete graphs, Ars Math. Contemp. 5 (2012), 159–173.

    MathSciNet  MATH  Google Scholar 

  6. [6]

    W. Imrich, J. Jerebic and S. Klavžar: The distinguishing number of Cartesian products of complete graphs, European J. Combin. 29 (2008), 922–929.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    W. Imrich, R. Kalinowski, F. Lehner and M. Pilśniak: Endomorphism breaking in graphs, Electron. J. Comb. 21, Research Paper P1.16, 2014.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    W. Imrich and S. Klavžar: Distinguishing Cartesian powers of graphs, J. Graph Theory 53 (2006), 250–260.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    W. Imrich, S. Klavžar and V. Trofimov: Distinguishing infinite graphs, J. Algebr. Comb. 41 (2015), 109–122.

    Article  MATH  Google Scholar 

  10. [10]

    W. Imrich, S. M. Smith, T. Tucker and M. E. Watkins: Infinite motion and 2-distinguishability of groups and graphs, preprint.

  11. [11]

    S. Klavžear, T.-L. Wong and X. Zhu: Distinguishing labellings of group action on vector spaces and graphs, J. Algebra 303 (2006), 626–641.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    S. Klavžear and X. Zhu: Cartesian powers of graphs can be distinguished by two labels, European J. Combin. 28 (2007), 303–310.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    C. Laflamme, L. Nguyen Van ThÉ and N. Sauer: Distinguishing number of count-able homogeneous relational structures, Electron. J. Combin. 17 Research Paper 20, 2010.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    F. Lehner: Random colorings and automorphism breaking in locally fintfinite graphs, Combinatorics, Probability and Computing 22 (2013), 885–909.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    A. Russell and R. Sundaram: A note on the asymptotics and computational complexity of graph distinguish ability, Electron. J. Combin. 5, Research Paper 23, 1998.

    MathSciNet  MATH  Google Scholar 

  16. [16]

    S. M. Smith, T. W. Tucker and M. E. Watkins: Distinguishability of infinite groups and graphs, Electron. J. Combin. 19, Research Paper 27, 2012.

    MathSciNet  MATH  Google Scholar 

  17. [17]

    T. W. Tucker: Distinguishing maps, Electron. J. Combin. 18 Paper 50, 2011.

    MathSciNet  MATH  Google Scholar 

  18. [18]

    M. E. Watkins and X. Zhou: Distinguishability of locally finite trees, Electron. J. Combin. 14, Research Paper 29, 2007.

    MathSciNet  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Florian Lehner.

Additional information

The author acknowledges the support of the Austrian Science Fund (FWF), project W1230-N13.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lehner, F. Distinguishing graphs with intermediate growth. Combinatorica 36, 333–347 (2016). https://doi.org/10.1007/s00493-015-3071-5

Download citation

Mathematics Subject Classification

  • 05C25
  • 05C15
  • 20B27