Advertisement

Designs, Codes and Cryptography

, Volume 65, Issue 3, pp 187–197 | Cite as

On the chromatic number of q-Kneser graphs

  • A. Blokhuis
  • A. E. Brouwer
  • T. Szőnyi
Open Access
Article

Abstract

We show that the q-Kneser graph qK 2k:k (the graph on the k-subspaces of a 2k-space over GF(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number q k  + q k−1 for k = 3 and for k < q log qq. We obtain detailed results on maximal cocliques for k = 3.

Keywords

Chromatic number q-analog of Kneser graph 

Mathematics Subject Classification (2000)

51E20 05B25 05D99 

Notes

Acknowledgments

Author A. Blokhuis acknowledges support from ERC grant DISCRETECONT 227701. This research began when T. Szőnyi was visiting Eindhoven University of Technology. The hospitality of TU/e and the financial support of EIDMA and Lex Schrijver’s Spinoza grant is gratefully acknowledged. Later T. Szőnyi was partly supported by OTKA Grant K 81310. In Spring 2004 the first author visited Günther Ziegler in Berlin. One of the problems he suggested to look at was determining the chromatic number of the q-Kneser graph.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Blokhuis A., Brouwer A.E., Chowdhury A., Frankl P., Patkós B., Mussche T., Szőnyi T.: A Hilton-Milner theorem for vector spaces. Electr. J. Combin. 17, R71 (2010)Google Scholar
  2. 2.
    Eisfeld J., Storme L., Sziklai P.: Minimal covers of the Klein quadric. J. Combin. Theory Ser. A 95, 145–157 (2001)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Frankl P., Wilson R.M.: The Erdős-Ko-Rado theorem for vector spaces. J. Combin. Theory Ser. A 43, 228–236 (1986)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Godsil C.D., Newman M.W.: Independent sets in association schemes. Combinatorica. 26, 431–443 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hsieh W.N.: Intersection theorems for systems of finite vector spaces. Discrete Math. 12, 1–16 (1975)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Mussche T.: Extremal combinatorics in generalized Kneser graphs, Ph.D. Thesis, Eindhoven University of Technology (2009).Google Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Dept. of MathematicsEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Institute of MathematicsEötvös Loránd UniversityBudapestHungary
  3. 3.Computer and Automation Research InstituteHungarian Academy of SciencesBudapestHungary

Personalised recommendations