The Chromatic Number of Two Families of Generalized Kneser Graphs Related to Finite Generalized Quadrangles and Finite Projective 3-Spaces

Klaus Metsch

doi:10.37236/10239

Abstract

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective space $\mathrm{PG}(3,q)$ 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 $q\geqslant 43$ 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 $q\geqslant 47$, 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.

Highlights

A chamber of a projective 3-space is a set consisting of a point P, a line and a plane π that are mutually incident
We are interested in the graph whose vertices are chambers of a three dimensional projective space of finite order q where two vertices are adjacent, if the corresponding chambers are opposite
Let Γ be the graph whose vertices are the chambers of a projective 3-space of finite order q with two vertices being adjacent when the corresponding chambers are in general position

Summary

Introduction

A chamber of a projective 3-space is a set consisting of a point P , a line and a plane π that are mutually incident. We are interested in the graph whose vertices are chambers of a three dimensional projective space of finite order q where two vertices are adjacent, if the corresponding chambers are opposite It was shown in [6] that the independence number of this graph is (q2 + q + 1)(q + 1). For many sets S with |S| > 1, the independence number was determined in [6] but neither the chromatic number nor the structure of the largest independent sets is known This applies to the graph defined above whose vertices are chambers of a projective 3-space of order q. Consider the graph Γ obtained from a generalized quadrangle of order (2, 2) embedded in PG(4, 2) and consisting of the points and lines of a parabolic quadric Q(4, 2) of PG(4, 2).

Generalized quadrangles

A stability result

The chromatic number