Random Colourings and Automorphism Breaking in Locally Finite Graphs

A colouring of a graph G is called distinguishing if its stabiliser in AutG 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 AutG 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. MSC 2010: 05E18, 20B27, 05C63.