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arXiv · 1611.02063

A note on Reed's Conjecture about $\omega$, $\Delta$ and $\chi$ with respect to vertices of high degree

Vera Weil
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Abstract

Reed conjectured that for every graph, $\chi \leq \left \lceil \frac{\Delta + \omega + 1}{2} \right \rceil$ holds, where $\chi$, $\omega$ and $\Delta$ denote the chromatic number, clique number and maximum degree of the graph, respectively. We develop an algorithm which takes a hypothetical counterexample as input. The output discloses some hidden structures closely related to high vertex degrees. Consequently, we deduce two graph classes where Reed's Conjecture holds: One contains all graphs in which the vertices of degree at least $5$ form a stable set. The other contains all graphs in which every induced cycle of odd length contains a vertex of at most degree 3.

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Vera Weil. 2016-11-07. A note on Reed's Conjecture about $\omega$, $\Delta$ and $\chi$ with respect to vertices of high degree. https://arxiv.org/abs/1611.02063

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