The reverse mathematics of Brooks' theorem

TL;DR

This paper analyzes Brooks' theorem within reverse mathematics, showing how its proof strength varies with graph degree restrictions, and establishes equivalences with subsystems like RCA_0 and WKL_0.

Contribution

It characterizes the logical strength of Brooks' theorem and its variants in the framework of reverse mathematics, revealing their equivalences to known subsystems.

Findings

Restricted Brooks' theorem for degree ≥3 provable in RCA_0

General Brooks' theorem equivalent to WKL_0 over RCA_0

Degree 2 Brooks' theorem also equivalent to WKL_0

Abstract

This is an analysis of the status of Brooks' Theorem, a celebrated result in graph coloring, from the point of view of Reverse Mathematics. We prove that the restriction of Brooks' theorem to bounded graphs of degree greater than or equal to $3$ is provable in $\mathsf{RCA}_0$, while the statement for arbitrary graphs is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. Brooks' Theorem for degree $2$, even when restricted to bounded graphs, is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$.