An improvement on Brooks' Theorem

We prove that $χ(G) \leq \max {ω(G), Δ_2(G), (5/6)(Δ(G) + 1)}$ for every graph $G$ with $Δ(G) \geq 3$. Here $Δ_2$ 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.

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