Random colorings and automorphism breaking in locally finite graphs

TL;DR

This paper investigates the properties of random 2-colourings in locally finite graphs, showing that such colourings almost surely break all automorphisms in various classes, supporting conjectures about automorphism fixing.

Contribution

It demonstrates that random 2-colourings are almost surely distinguishing in several classes of locally finite graphs, providing probabilistic evidence for the automorphism breaking conjecture.

Findings

Stabiliser of random 2-colourings is almost surely nowhere dense.

In certain classes, random 2-colourings are almost surely distinguishing.

Supports conjecture that infinite automorphisms can be broken by 2-colourings.

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.