On the difference between the chromatic and cochromatic number
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916362559946752 |
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| author | Steiner, Raphael |
| author_facet | Steiner, Raphael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the difference between the chromatic and cochromatic number Steiner, Raphael Combinatorics 05C15, 05C69 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. |
| title | On the difference between the chromatic and cochromatic number |
| topic | Combinatorics 05C15, 05C69 |
| url | https://arxiv.org/abs/2408.02400 |