Coloring $\Delta$-Critical Graphs With Small High Vertex Cliques

TL;DR

This paper proves a unique characterization of certain critical graphs with high chromatic number and degree, confirming a conjecture when a specific subgraph has a clique number of one.

Contribution

It establishes that the only critical graphs with specified degree and clique conditions are complete graphs, confirming a conjecture for the case when the high-degree subgraph is an independent set.

Findings

Proves the uniqueness of critical graphs with given degree and clique constraints.

Confirms a conjecture of Kierstead and Kostochka for a special case.

Identifies the structure of critical graphs with small high vertex cliques.

Abstract

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