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

TL;DR

This paper establishes near-optimal upper bounds on the chromatic number of Erd ext{"o}s-Rényi orthogonal polarity graphs for certain prime powers, and identifies small subgraphs with low chromatic number.

Contribution

It provides new upper bounds on the chromatic number of ER_q graphs for even and odd prime powers, improving previous results and analyzing subgraph structures.

Findings

For even powers of odd primes, hi(ER_q)  2b7a0 + O(b7a0 / a0log q)

Bounds are tight up to a factor of 2 for these graphs

Existence of small subgraphs with chromatic number at most 3 for large q

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.