Breaking graph symmetries by edge colourings

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colours needed in an edge colouring which is not preserved by any non-trivial automorphism. Broere and Pil\'sniak conjectured that if every non-trivial automorphism of a countable graph $G$ moves infinitely many edges, then $D'(G) \leq 2$. We prove this conjecture.

[1]  Rögnvaldur G. Möller,et al.  Local Finiteness, Distinguishing Numbers, and Tucker's Conjecture , 2015, Electron. J. Comb..

[2]  Wilfried Imrich,et al.  Distinguishing graphs with infinite motion and nonlinear growth , 2014, Ars Math. Contemp..

[3]  Wilfried Imrich,et al.  Bounds for Distinguishing Invariants of Infinite Graphs , 2017, Electron. J. Comb..

[4]  Wilfried Imrich,et al.  Infinite motion and 2-distinguishability of graphs and groups , 2013 .

[5]  Florian Lehner.  Distinguishing graphs with intermediate growth , 2016, Comb..

[6]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[7]  Xiangqian Zhou,et al.  Distinguishability of Locally Finite Trees , 2007, Electron. J. Comb..

[8]  Thomas W. Tucker,et al.  Distinguishability of Infinite Groups and Graphs , 2012, Electron. J. Comb..

[9]  Florian Lehner.  Random Colourings and Automorphism Breaking in Locally Finite Graphs , 2013, Comb. Probab. Comput..

[10]  Wilfried Imrich,et al.  Distinguishing Infinite Graphs , 2007, Electron. J. Comb..

[11]  Alexander Russell,et al.  A Note on the Asymptotics and Computational Complexity of Graph Distinguishability , 1998, Electron. J. Comb..

[12]  Monika Pilsniak,et al.  Distinguishing graphs by edge-colourings , 2015, Eur. J. Comb..

[13]  Michael O. Albertson,et al.  Symmetry Breaking in Graphs , 1996, Electron. J. Comb..

[14]  H. Jung,et al.  Zu einem Isomorphiesatz von H. Whitney für Graphen , 1966 .

[15]  Izak Broere,et al.  The Distinguishing Index of Infinite Graphs , 2015, Electron. J. Comb..

[16]  Thomas W. Tucker,et al.  Distinguishing Maps , 2011, Electron. J. Comb..