Distinguishing graphs with intermediate growth

Abstract

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.

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References

  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 

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Correspondence to Florian Lehner.

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

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

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Mathematics Subject Classification

  • 05C25
  • 05C15
  • 20B27