Let G be a triangle-free graph with maximum degree ∆(G). We show that the chromatic number χ(G) is less than 67(1 + o(1))∆/ log∆.

The well-known Brooks’ Theorem says that each graph G of maximum degree k ≥ 3 is k-colorable unless G = Kk+1. We generalize this theorem by allowing higher degree vertices with prescribed types of neighborhood.

We give a proof of Brooks’ theorem as well as its list coloring extension using the algebraic method of Alon and Tarsi.

We prove that if G is the line graph of a multigraph, then the chromatic number χ(G) of G is at most max {

Brooks and colleagues (S. W. Allen & L. R. Brooks, 1991; G. Regehr & L. R. Brooks, 1993) have shown that the classification of transfer stimuli is influenced by their similarity to training stimuli, even when a perfect classification rule is available. It is argued that the original effect obtained by Brooks and colleagues might have resulted from two potential confounding variables. Once these...

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 ...