On the Chromatic Number of some generalized Kneser Graphs

TL;DR

This paper determines the chromatic number of certain generalized Kneser graphs related to flags of vector spaces over finite fields for large q, and characterizes the colorings that achieve this bound.

Contribution

It provides the exact chromatic number for specific flag-based Kneser graphs and extends the result to a broader class of such graphs for large q.

Findings

Chromatic number of qΓ_{7,{3,4}} determined for large q

Characterization of colorings attaining the bound

Generalization to qΓ_{2d+1,{d,d+1}} for large q

Abstract

We determine the chromatic number of the Kneser graph q{\Gamma}_{7,{3,4}} of flags of vectorial type {3, 4} of a rank 7 vector space over the finite field GF(q) for large q and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all large independent sets. Furthermore, our proof is more general in the following sense: it provides the chromatic number of the Kneser graphs q{\Gamma}_{2d+1,{d,d+1}} of flags of vectorial type {d, d+1} of a rank 2d+1 vector space over GF(q) for large q as long as the large independent sets of the graphs are only the ones that are known.