On the chromatic number of some generalized Kneser graphs

Abstract

We determine the chromatic number of the Kneser graph $q{\Gamma }_{7,\left\{3,4\right\}}$ of flags of vectorial type $\left\{3,4\right\}$ of a rank 7 vector space over the finite field $\mathrm{GF}\left(q\right)$ 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 $q{\Gamma }_{2d+1,\left\{d,d+1\right\}}$ of flags of vectorial type $\left\{d,d+1\right\}$ of a rank $2d+1$ vector space over $\mathrm{GF}\left(q\right)$ for large $q$ as long as the large independent sets of the graphs are only the ones that are known.