The colouring number of infinite graphs

TL;DR

This paper characterizes the coloring number of infinite graphs through forbidden subgraphs and demonstrates that graphs with infinite coloring number admit a well-ordering that reflects both their coloring number and size.

Contribution

It provides a new characterization of the coloring number of infinite graphs using forbidden subgraphs and establishes the existence of a well-ordering that witnesses both the coloring number and the cardinality.

Findings

Characterization of infinite graph coloring number via forbidden subgraphs

Existence of a well-ordering witnessing coloring number and cardinality

Applicable to graphs with any infinite cardinality

Abstract

We show that, given an infinite cardinal $\mu$, a graph has colouring number at most $\mu$ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite colouring number has a well-ordering of its vertices that simultaneously witnesses its colouring number and its cardinality.