Distinguishing colorings, proper colorings, and covering properties without the Axiom of Choice

TL;DR

This paper explores the equivalence of various graph coloring and covering properties to Kőnig's Lemma within ZF set theory, without relying on the Axiom of Choice, and introduces new conditions for graph properties.

Contribution

It demonstrates the equivalence of several graph properties to Kőnig's Lemma in ZF and extends previous results by removing the assumption that color sets are well-orderable.

Findings

Equivalence of graph coloring properties to Kőnig's Lemma in ZF

New conditions for proper coloring, edge cover, matching, and dominating sets in graphs

Minimal dominating set existence under Axiom of Choice for 2-element families

Abstract

We work with simple graphs in ZF (Zermelo--Fraenkel set theory without the Axiom of Choice (AC)) and assume that the sets of colors can be either well-orderable or non-well-orderable to prove that the following statements are equivalent to K\H{o}nig Lemma: (a) Any infinite locally finite connected graph G such that the minimum degree of G is greater than k, has a chromatic number for any fixed integer k greater than or equal to 2. (b) Any infinite locally finite connected graph has a chromatic index. (c) Any infinite locally finite connected graph has a distinguishing number. (d) Any infinite locally finite connected graph has a distinguishing index. Our results strengthen some results of Stawiski from a recent paper on the role of the Axiom of Choice in proper and distinguishing colorings since he assumed that the sets of colors can be well-ordered. We also formulate new conditions for the existence of irreducible proper coloring, minimal edge cover, maximal matching, and minimal dominating set in connected bipartite graphs and locally finite connected graphs, which are either equivalent to AC or K\H{o}nig Lemma. Moreover, we show that if the Axiom of Choice for families of 2 element sets holds, then the Shelah--Soifer graph has a minimal dominating set.