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

Landon Rabern

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

Combinatorics 2011-02-08 v1

Authors: Landon Rabern

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.

Keywords

hadwiger conjecture graph coloring graph theory

Cite

@article{arxiv.1102.1023,
  title  = {Coloring $\Delta$-Critical Graphs With Small High Vertex Cliques},
  author = {Landon Rabern},
  journal= {arXiv preprint arXiv:1102.1023},
  year   = {2011}
}

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