The role of the Axiom of Choice in proper and distinguishing colourings

Abstract

Call a colouring of a graph \emph{distinguishing} if the only automorphism which preserves it is the identity. We investigate the role of the Axiom of Choice in the existence of certain proper or distinguishing colourings in both vertex and edge variants with emphasis on locally finite connected graphs. In particular, we show that every locally finite connected graph has a distinguishing or proper colouring if and only if K\H{o}nig's Lemma holds. We show that we cannot prove in ZF that such colourings exist even for connected graphs with maximum degree 3. We also formulate few conditions about distinguishing and proper colouring which are equivalent to the Axiom of Choice.