Graphs with χ=Δ Have Big Cliques

Brooks' theorem implies that if a graph has $\Delta\ge 3$ and $\chi > \Delta$, then $\omega=\Delta+1$. Borodin and Kostochka conjectured that if $\Delta\ge 9$ and $\chi\ge \Delta$, then $\omega\ge \Delta$. We show that if $\Delta\ge 13$ and $\chi\ge \Delta$, then $\omega \ge \Delta-3$. For a graph $G$, let ${\mathcal{H}(G)}$ denote the subgraph of $G$ induced by vertices of degree $\Delta$. We also show that if $\chi\ge \Delta$, then $\omega\ge \Delta$ or $\omega({\mathcal{H}(G)})\ge \Delta-5$.

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