Random colorings and automorphism breaking in locally finite graphs
Combinatorics
2013-04-25 v1
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.
Cite
@article{arxiv.1304.6642,
title = {Random colorings and automorphism breaking in locally finite graphs},
author = {Florian Lehner},
journal= {arXiv preprint arXiv:1304.6642},
year = {2013}
}