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The reverse mathematics of Brooks' theorem

Alberto Marcone, Gian Marco Osso

TL;DR

The paper investigates the logical strength of Brooks' theorem within reverse mathematics, distinguishing bounded-graph cases from the general theorem. It shows that the full Brooks' theorem is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$, while the bounded-degree version with $\Delta(G) \ge 3$ is provable in $\mathsf{RCA}_0$; the $\Delta(G)=2$ case for bounded graphs is also equivalent to $\mathsf{WKL}_0$. A key result is that $\mathsf{BT}^{\mathsf{b}}_2$ is equivalent to $\mathsf{WKL}_0$, and that $\mathsf{RCA}_0$ proves $\mathsf{BT}^{\mathsf{b}}_3$ using circle-trees and a Tverberg-style construction, with an induction to extend to all $d \ge 3$. The analysis ties graph-coloring results to compactness principles via the De Bruijn–Erdős theorem and introduces circle-tree structures, germs, and $P$/$Q$-vertices to build effective colorings, highlighting the boundary between constructive colorings and compactness arguments. Overall, the work clarifies the precise axioms needed for Brooks' theorem and reveals Weihrauch-reduction compatible uniform proofs for the bounded-case variants.

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 $\mathsf{RCA}_0$.

The reverse mathematics of Brooks' theorem

TL;DR

The paper investigates the logical strength of Brooks' theorem within reverse mathematics, distinguishing bounded-graph cases from the general theorem. It shows that the full Brooks' theorem is equivalent to over , while the bounded-degree version with is provable in ; the case for bounded graphs is also equivalent to . A key result is that is equivalent to , and that proves using circle-trees and a Tverberg-style construction, with an induction to extend to all . The analysis ties graph-coloring results to compactness principles via the De Bruijn–Erdős theorem and introduces circle-tree structures, germs, and /-vertices to build effective colorings, highlighting the boundary between constructive colorings and compactness arguments. Overall, the work clarifies the precise axioms needed for Brooks' theorem and reveals Weihrauch-reduction compatible uniform proofs for the bounded-case variants.

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 is provable in , while the statement for arbitrary graphs is equivalent to over . Brooks' Theorem for degree , even when restricted to bounded graphs, is equivalent to over .
Paper Structure (13 sections, 29 theorems, 4 equations, 6 figures)

This paper contains 13 sections, 29 theorems, 4 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Basic graph theoretic notions
  4. Graphs in $\mathsf{RCA}_0$
  5. The strength of the full Brooks' Theorem
  6. Brooks' Theorem for bounded graphs
  7. The strength of $\mathsf{BT}^{\mathsf{b}}_2$
  8. $\mathsf{RCA}_0$ proves $\mathsf{BT}^{\mathsf{b}}_3$
  9. Circle-trees
  10. $P$-vertices, $Q$-vertices, germs, and circle-trees in graphs with degree $3$
  11. Tverberg's proof
  12. $\mathsf{RCA}_0$ proves $\forall d \geq 3 \,\, \mathsf{BT}^{\mathsf{b}}_d$.
  13. On Schmerl's effective proof of Brooks' Theorem

Key Result

Theorem 1.1

Let $d \geq 2$ and let $G$ be a graph such that all nodes in $G$ have at most $d$ neighbors. Suppose that $G$ does not have any $d+1$-clique as a subgraph. If $d=2$, additionally assume that $G$ does not have any odd cycles. Then $G$ is $d$-colorable.

Figures (6)

  • Figure 1: A portion of the graph $H_{2}$ for a pair of functions $f$ and $g$ such that $n= f(m)$, $k = g(m)$, and $i \notin \mathop{\mathrm{ran}}\nolimits(f) \cup \mathop{\mathrm{ran}}\nolimits(g)$.
  • Figure 2: A portion of the graph $H_{3}$ for a pair of functions $f$ and $g$ such that $f(m)=n$, $g(m)=k$, $i \notin \mathop{\mathrm{ran}}\nolimits(f) \cup \mathop{\mathrm{ran}}\nolimits(g)$.
  • Figure 3: A portion of the graph $G$ corresponding to functions $f$, $g$ such that $0 \notin \mathop{\mathrm{ran}}\nolimits(f {\upharpoonright} 4) \cup \mathop{\mathrm{ran}}\nolimits(g{\upharpoonright} 4)$, $1=f(3)$ and $2=g(2)$.
  • Figure 6: The third case of an application of the rule \ref{['it:gt']}.
  • Figure 7: Subcase in which the deletion of $x$ does not lead to the creation of germs with priority over the germ $aEbE\dots Ey$.
  • ...and 1 more figures

Theorems & Definitions (94)

  • Theorem 1.1: Brooks, Brooks
  • Theorem 1.2: De Bruijn-Erdős
  • Definition 2.1: Graph
  • Definition 2.2: Degree of a vertex
  • Definition 2.3: Graphs of finite degree
  • Definition 2.4: Neighborhoods
  • Definition 2.5: Induced subgraphs
  • Definition 2.6: Cliques
  • Definition 2.7: Paths and cycles
  • Definition 2.8: Connected components, connectedness
  • ...and 84 more