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arXiv · 1408.4065

On the chromatic number of the Erd\H{o}s-R\'enyi orthogonal polarity graph

Xing Peng·Mike Tait·Craig Timmons
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Abstract

For a prime power $q$, let $ER_q$ denote the Erd\H{o}s-R\'enyi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $\chi ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best possible up to a constant factor of at most 2. If $q$ is an odd power of an odd prime and satisfies some condition on irreducible polynomials, then we improve the best known upper bound for $\chi(ER_{q})$ substantially. We also show that for sufficiently large $q$, every $ER_q$ contains a subgraph that is not 3-chromatic and has at most 36 vertices.

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Xing Peng, Mike Tait, Craig Timmons. 2014-08-18. On the chromatic number of the Erd\H{o}s-R\'enyi orthogonal polarity graph. https://arxiv.org/abs/1408.4065

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