Brooks' type theorems for coloring parameters of locally finite graphs and Konig's Lemma

TL;DR

This paper explores the set-theoretic foundations of graph coloring theorems for infinite graphs, demonstrating their dependence on the Axiom of Choice and Konig's Lemma, and establishing new conditions for various coloring parameters.

Contribution

It proves that certain Brooks-type theorems for infinite graphs are not provable in ZF and links their validity to Konig's Lemma, strengthening recent results.

Findings

Brooks' theorems for infinite graphs depend on Konig's Lemma.

New ZF conditions for various coloring parameters are established.

Upper bounds for list-distinguishing chromatic number in ZFC are provided.

Abstract

In the past, analogues to Brooks' theorem have been found for various parameters of graph coloring for infinite locally finite connected graphs in ZFC. We prove these theorems are not provable in ZF (i.e. the Zermelo-Fraenkel set theory without the Axiom of Choice (AC)). Moreover, such theorems follow from Konig's Lemma (every infinite locally finite connected graph has a ray-a weak form of AC) in ZF. In ZF, we formulate new conditions for the existence of the distinguishing chromatic number, the distinguishing chromatic index, the total chromatic number, the total distinguishing chromatic number, the odd chromatic number, and the neighbor-distinguishing index in infinite locally finite connected graphs, which are equivalent to Konig's Lemma. In this direction, we strengthen a recent result of Stawiski from 2023. We also figured out the upper bound for list-distinguishing chromatic number for infinite graphs in ZFC (i.e. the Zermelo-Fraenkel set theory with the Axiom of Choice (AC)).