Cranston–Rabern conjecture on maximum-degree cliques

Let $G$ be a graph. Write $\Delta$ for its maximum degree, $\omega$ for its clique number, and $\chi$ for its chromatic number. Let $\mathcal{H}(G)$ be the subgraph induced by the vertices of degree $\Delta$. Cranston–Rabern conjecture. If $\chi\ge\Delta$, then

$$\omega\ge\Delta\quad\text{or}\quad\omega(\mathcal{H}(G))\ge\Delta-4.$$

The conjecture strengthens the paper’s main theorem by requiring a larger clique among the maximum-degree vertices. It is proved for $\Delta=6$ and $\Delta=7$, and for $\Delta=3k+1$ with $k\ge3$; the stated bound is expected to be tight because of the graph $O_5$.