Israel Journal of Mathematics

, Volume 170, Issue 1, pp 125–134

A short proof of w1n (Hom(C2r+1, Kn+2)) = 0 for all n and A graph colouring theorem by Babson and Kozlov

Article

Abstract

We show that the n-th power of the first Stiefel-Whitney class of the ℤ2-action on the graph complex Hom(C2r+1, Kn+2) is zero, confirming a conjecture by Babson and Kozlov. This yields a considerably simplified proof of their graph colouring theorem, which is also known as the Lovsz conjecture.

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References

  1. [BK03]
    E. Babson and D. N. Kozlov, Topological obstructions to graph colorings, Electronic Research Announcements of the American Mathematical Society 9 (2003), 61–68.MATHCrossRefMathSciNetGoogle Scholar
  2. [BK06]
    E. Babson and D. N. Kozlov, Complexes of graph homomorphisms, Israel Journal of Mathematics 152 (2006), 285–312.MATHCrossRefMathSciNetGoogle Scholar
  3. [BK07]
    E. Babson and D. N. Kozlov, Proof of the Lovász conjecture, Annals of Mathematics 165 (2007), 965–1007.MATHCrossRefMathSciNetGoogle Scholar
  4. [CL06]
    P. Csorba and F. H. Lutz, Graph coloring manifolds, Contemporary Mathematics 423 (2006), 51–69.MathSciNetGoogle Scholar
  5. [Cso05]
    P. Csorba, Non-tidy Spaces and Graph Colorings, PhD thesis, ETH Zurich, 2005.Google Scholar
  6. [Koz06]
    D. N. Kozlov, Cobounding odd cycle colorings, Electronic Research Announcements of the AMS 12 (2006), 53–55 (electronic). ISSN 1079-6762.MATHCrossRefGoogle Scholar
  7. [Koz07]
    D. N. Kozlov, Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes, in Geometric Combinatorics, (Miller, E., Reiner, V., and Sturmfels, B., eds.), IAS/Park City Math. Ser., vol. 13, American Mathematical Society, Providence, RI, 2007, pp. 249–315.Google Scholar
  8. [Lov78]
    L. Lovász, Kneser’s conjecture, chromatic number and homotopy, Journal of Combinatorial Theory. Series A 25 (1978), 319–324.MATHCrossRefMathSciNetGoogle Scholar
  9. [Sch06]
    C. Schultz, Graph colourings, spaces of edges and spaces of circuits, Advances in Mathematics, 2006, submitted. math.CO/0606763.Google Scholar
  10. [Sch05]
    C. Schultz, Small models of graph colouring manifolds and the Stiefel manifolds Hom(C 5, K n), Journal of Combinatorial Theory, Series A, 115 (2008), 84–104. in press, 21 pp.MATHCrossRefMathSciNetGoogle Scholar
  11. [Wal88]
    J. W. Walker, Canonical homeomorphisms of posets European Journal of Combinatorics 9(2) (1988), 97–107. ISSN 0195-6698.MATHMathSciNetGoogle Scholar
  12. [Živ05]
    R. T. Živaljević, Combinatorial groupoids, cubical complexes, and the Lovász conjecture, 2005, 28 pp. math.CO/0510204Google Scholar

Copyright information

© Hebrew University Magnes Press 2009

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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