Coloring $Δ$-Critical Graphs With Small High Vertex Cliques

We prove that $K_{χ(G)}$ is the only critical graph $G$ with $χ(G) \geq Δ(G) \geq 6$ and $ω(\mathcal{H}(G)) \leq \left \lfloor \frac{Δ(G)}{2} \right \rfloor - 2$. Here $\mathcal{H}(G)$ is the subgraph of $G$ induced on the vertices of degree at least $χ(G)$. Setting $ω(\mathcal{H}(G)) = 1$ proves a conjecture of Kierstead and Kostochka.

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