Geodesic
The chromatic number of two families of generalized Kneser graphs related to finite generalized quadrangles and finite projective 3-spaces
The electronic journal of combinatorics, Tome 28 (2021) no. 3
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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.
DOI : 10.37236/10239
Classification : 51E20, 05B25, 51E12, 05C10
Mots-clés : chromatic number, generalized Kneser graphs, finite generalized quadrangles, finite projective 3-spaces 
@article{10_37236_10239,
     author = {Klaus Metsch},
     title = {The chromatic number of two families of generalized {Kneser} graphs related to finite generalized quadrangles and finite projective 3-spaces},
     journal = {The electronic journal of combinatorics},
     year = {2021},
     volume = {28},
     number = {3},
     doi = {10.37236/10239},
     zbl = {1467.51006},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10239/}
}
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Klaus Metsch. The chromatic number of two families of generalized Kneser graphs related to finite generalized quadrangles and finite projective 3-spaces. The electronic journal of combinatorics, Tome 28 (2021) no. 3. doi: 10.37236/10239

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