Brooks' Comet

Brooks' Theorem and Beyond

We collect some of our favorite proofs of Brooks’ Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon–Tarsi orientations, since analo...

Books in Running Brooks

No Quantum Brooks' Theorem

First, I introduce quantum graph theory. I also discuss a known lower bound on the independence numbers and derive from it an upper bound on the chromatic numbers of quantum graphs. Then, I construct a family of quantum graphs that can be described as tropical, cyclical, and commutative. I also define a step logarithm function and express with it the bounds on quantum graph invariants in closed...

Brooks’ theorem for Bernoulli shifts

If Γ is an infinite group with finite symmetric generating set S, we consider the graph G(Γ, S) on [0, 1]Γ by relating two distinct points if an element of s sends one to the other via the shift action. We show that, aside from the cases Γ = Z and Γ = (Z/2Z) ∗ (Z/2Z), G(Γ, S) satisfies a measure-theoretic version of Brooks’ theorem: there is a G(Γ, S)-invariant conull Borel set B ⊆ [0, 1]Γ and ...

An Improvement on Brooks’ Theorem

We prove that χ(G) ≤ max { ω(G),∆2(G), 5 6 (∆(G) + 1) } for every graph G with ∆(G) ≥ 3. Here ∆2 is the parameter introduced by Stacho that gives the largest degree that a vertex v can have subject to the condition that v is adjacent to a vertex whose degree is at least as large as its own. This upper bound generalizes both Brooks’ Theorem and the Ore-degree version of Brooks’ Theorem.