Irreducible distinguishing colourings and the Axiom of Choice

TL;DR

This paper explores the concept of irreducible distinguishing colourings in graphs, establishing their existence for all graphs and linking their existence to the Axiom of Choice.

Contribution

It introduces the notion of irreducible distinguishing colourings and proves their existence for all graphs, connecting these concepts to foundational set theory.

Findings

Every graph has an irreducible distinguishing vertex colouring.

Graphs without isolated edges or with at most one isolated vertex have irreducible distinguishing edge colourings.

Existence of such colourings for all connected graphs (not K2) is equivalent to the Axiom of Choice.

Abstract

We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two non-empty classes of colours to obtain a (proper) distinguishing colouring. We show that every graph has an irreducible (proper) distinguishing vertex colouring and that every graph without isolated edge and with at most one isolated vertex has an irreducible (proper) distinguishing edge colouring. Moreover, we show that the existence of any of these colourings for every connected graph (not isomorphic to $K_2$) is equivalent to the Axiom of Choice.