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Random Colourings and Automorphism Breaking in Locally Finite Graphs

Published online by Cambridge University Press:  16 September 2013

Institute of Geometry, TU Graz, Kopernikusgasse 24, A 8010 Graz, Austria (e-mail:


A colouring of a graph G is called distinguishing if its stabilizer in Aut G is trivial. It has been conjectured that, if every automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We study properties of random 2-colourings of locally finite graphs and show that the stabilizer of such a colouring is almost surely nowhere dense in Aut G and a null set with respect to the Haar measure on the automorphism group. We also investigate random 2-colourings in several classes of locally finite graphs where the existence of a distinguishing 2-colouring has already been established. It turns out that in all of these cases a random 2-colouring is almost surely distinguishing.

Copyright © Cambridge University Press 2013 

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