Schrijver Graphs and Projective Quadrangulations



In a recent paper, Kaiser and Stehlík (J Combin Theory Ser B 113:1–17, 2015) have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the n-dimensional projective space \(\mathbb{P}^{n}\) is at least (n + 2)-chromatic, unless it is bipartite. They conjectured that for any integers k ≥ 1 and n ≥ 2k + 1, the Schrijver graph SG(n, k) contains a spanning subgraph which is a non-bipartite quadrangulation of \(\mathbb{P}^{n-2k}\). The purpose of this paper is to prove the conjecture.



We thank two anonymous reviewers for a careful reading of the first versions and a number of suggestions. This project was started while the second author was visiting Fabricio Benevides and Víctor Campos at the Universidade Federal do Ceará.


  1. 1.
    D. Attali, A. Lieutier, D. Salinas, Efficient data structure for representing and simplifying simplicial complexes in high dimensions. Int. J. Comput. Geom. Appl. 22(4), 279–303 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    J. Gimbel, C. Thomassen, Coloring graphs with fixed genus and girth. Trans. Am. Math. Soc. 349, 4555–4564 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)MATHGoogle Scholar
  4. 4.
    T. Kaiser, M. Stehlík, Colouring quadrangulations of projective spaces. J. Combin. Theory Ser. B 113, 1–17 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    M. Kneser, Aufgabe 300. Jahresber. Deutsch. Math.-Verein. 58, 27 (1955)Google Scholar
  6. 6.
    L. Lovász, Kneser’s conjecture, chromatic number, and homotopy. J. Combin. Theory Ser. A 25(3), 319–324 (1978)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Matoušek, Using the Borsuk-Ulam Theorem. Universitext (Springer, Berlin, 2003)MATHGoogle Scholar
  8. 8.
    J.R. Munkres, Elements of Algebraic Topology (Addison–Wesley Publishing Company, Menlo Park, 1984)MATHGoogle Scholar
  9. 9.
    E. Nevo, Higher minors and Van Kampen’s obstruction. Math. Scand. 101(2), 161–176 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    A. Schrijver, Vertex-critical subgraphs of Kneser graphs. Nieuw Arch. Wisk. (3) 26(3), 454–461 (1978)Google Scholar

Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Institute for Theoretical Computer Science (CE-ITI), and European Centre of Excellence NTIS (New Technologies for the Information Society)University of West BohemiaPilsenCzech Republic
  2. 2.Laboratoire G-SCOPUniversité Grenoble AlpesGrenobleFrance

Personalised recommendations