Geodesic
The role of the Axiom of Choice in proper and distinguishing colourings
Ars Mathematica Contemporanea, Tome 23 (2023) no. 3, article no. 10, 8 p.

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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 Kőnig's Lemma holds. This statement holds for both vertex and edge colourings. Furthermore, we show that it is improvable in ZF that such colourings exist even for every connected graph with maximum degree 3.We also formulate few conditions about distinguishing and proper colourings which are equivalent to the Axiom of Choice.
DOI : 10.26493/1855-3974.2863.4b9
Keywords: Proper colourings, distinguishing colourings, asymmetric colourings, infinite graphs, graph automorphisms, Axiom of Choice 
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Marcin Stawiski. The role of the Axiom of Choice in proper and distinguishing colourings. Ars Mathematica Contemporanea, Tome 23 (2023) no. 3, article  no. 10, 8 p. doi : 10.26493/1855-3974.2863.4b9. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2863.4b9/

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