The chromatic number of a two families of generalized Kneser graphs related to finite generalized quadrangles and finite projective 3-spaces

TL;DR

This paper determines the chromatic number of a graph derived from chambers in finite projective spaces and generalized quadrangles, revealing precise bounds and structure for large q values.

Contribution

It provides the first exact determination of the chromatic number for these graphs and characterizes the size of maximal independent sets for large q.

Findings

Chromatic number of the graph is q^2 + q for q ≥ 47.

Largest independent sets have size (q^2+q+1)(q+1)^2 for q ≥ 43.

Maximal independent sets not largest are at most constant times q^3 in size.

Abstract

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective space $PG(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\ge 43$ we determine the largest independent set of $\Gamma$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\ge 47$, this information is then used to show that $\Gamma$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove similar results for the graph that is built in the same way on the chambers of the generalized quadrangle.