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arXiv · 1304.6642

Random colorings and automorphism breaking in locally finite graphs

Florian Lehner
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Abstract

A colouring of a graph G is called distinguishing if its stabiliser 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 stabiliser 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.

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Florian Lehner. 2013-04-24. Random colorings and automorphism breaking in locally finite graphs. https://arxiv.org/abs/1304.6642

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