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

TL;DR

This paper explores how the Axiom of Choice influences the existence of proper and distinguishing colourings in graphs, revealing that certain colourings are equivalent to AC and depend on Kőnig's Lemma.

Contribution

It establishes the equivalence between the existence of specific graph colourings and foundational set-theoretic principles like the Axiom of Choice and Kőnig's Lemma.

Findings

Existence of distinguishing colourings with countable colours is equivalent to Kőnig's Lemma.

Existence of such colourings cannot be proven in ZF for graphs with maximum degree 3.

Certain conditions on colourings are shown to be equivalent to the Axiom of Choice.

Abstract

Call a colouring of a graph 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 special emphasis on locally finite connected graphs. We show that every locally finite connected graph has a distinguishing colouring with at most countable number of colours or every locally finite connected graph has a proper colouring with at most countable number of colours if and only if K\H{o}nig's Lemma holds. This statement holds for both vertex and edge colourings. Furthermore, we show that it is not provable in ZF that such colourings exist even for every connected graph with maximum degree 3. We also formulate a few conditions about distinguishing and proper colourings which are equivalent to the Axiom of Choice.