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arXiv · 2408.02400

On the difference between the chromatic and cochromatic number

Raphael Steiner
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Abstract

The cochromatic number $ζ(G)$ of a graph $G$ is the smallest number of colors in a vertex-coloring of $G$ such that every color class forms an independent set or a clique. In three papers written around 1990, Erdős, Gimbel and collaborators raised several open problems regarding the relationship of the chromatic and cochromatic number of a graph. In this short note, we address several of these problems, in particular -we disprove a conjecture of Erdős, Gimbel and Straight from 1988, -answer negatively a problem posed by Erdős and Gimbel in 1993, and -give positive evidence for a 1000\$--question of Erdős and Gimbel.

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Raphael Steiner. 2024-08-20. On the difference between the chromatic and cochromatic number. https://arxiv.org/abs/2408.02400

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