Oriented matroids and Ky Fan’s theorem

Abstract

L. Lovász (Matroids and Sperner’s Lemma, Europ. J. Comb. 1 (1980), 65–66) has shown that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature. Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C m,r.

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References

  1. [1]

    A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler: Oriented Matroids, Encyclopedia of Mathematics, Vol. 46, Cambridge Univ. Press, Cambridge, 1993.

    Google Scholar 

  2. [2]

    G. E. Bredon: Topology and Geometry, Graduate Texts in Mathematics 139, Springer, 1995.

  3. [3]

    T. H. Brylawski and G. M. Ziegler: Topological representation of dual pairs of oriented matroids, Discrete Comput. Geom.10 (1993), 237–240.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    A. Dold: Lectures on Algebraic Topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer, 1972.

  5. [5]

    K. Fan: A generalization of Tucker’s combinatorial lemma with topological applications, Annals Math., II. Ser.56 (1952), 431–437.

    Article  Google Scholar 

  6. [6]

    B. Hanke, R. Sanyal, C. Schultz and G. M. Ziegler: Combinatorial Stokes formulas via minimal resolutions, J. Combinatorial Theory, Ser. A116(2) (2009), 404–420.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    D. Kozlov: Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics, Vol. 21, Springer, 2008.

  8. [8]

    S. Kryńsky: Remarks on matroids and Sperner’s lemma, Europ. J. Combinatorics11 (1990), 485–488.

    Google Scholar 

  9. [9]

    S.-H. Lee and M.-H. Shih: Sperner matroid, Arch. Math.81 (2003), 103–112.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    B. Lindström: On matroids and Sperner’s lemma, Europ. J. Combinatorics2 (1981), 65–66.

    Google Scholar 

  11. [11]

    M. de Longueville and R. T. Živaljević: The Borsuk-Ulam-property, Tuckerproperty and constructive proofs in combinatorics; J. Combinatorial Theory, Ser. A113 (2006), 839–850.

    MATH  Article  Google Scholar 

  12. [12]

    L. Lovász: Matroids and Sperner’s Lemma, Europ. J. Comb.1 (1980), 65–66.

    MATH  Google Scholar 

  13. [13]

    J. Matoušek: Using the Borsuk-Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry. Springer, 2003.

  14. [14]

    J. Matoušek: A combinatorial proof of Kneser’s conjecture, Combinatorica24(1) (2004), 163–170.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    F. Meunier: A ℤq-Fan formula, Preprint, 2005, http://www.enpc.fr/lvmt/frederic.meunier/.

  16. [16]

    F. Meunier: Sperner labellings: a combinatorial approach; J. Combinatorial Theory, Ser. A113 (2006), 1462–1475.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    J. Richter-Gebert and G. M. Ziegler: Oriented matroids; Chapter 6 in Handbook of Discrete and Computational Geometry (J. E. Goodman, J. O’Rourke eds.), second edition, Chapman & Hall/CRC, 2004.

  18. [18]

    K. S. Sarkaria: Tucker-Ky Fan colorings, Proc. Amer. Math. Soc., Vol. 110, no. 4, 1990.

  19. [19]

    G. Simonyi and G. Tardos: Local chromatic number, Ky Fan’s theorem, and circular colorings; Combinatorica26(5) (2006), 587–626.

    Article  MathSciNet  Google Scholar 

  20. [20]

    A. W. Tucker: Some topological properties of disk and sphere, Proc. First Canadian Math. Congress (Montreal, 1945), 285–309.

  21. [21]

    G. M. Ziegler: What is a complex matroid, Discrete Comput. Geom.10 (1993), 313–348.

    MATH  Article  MathSciNet  Google Scholar 

  22. [22]

    G. M. Ziegler: Lectures on Polytopes, Graduate Texts in Mathematics, Springer, 1995.

  23. [23]

    G. M. Ziegler: Generalized Kneser coloring theorems with combinatorial proofs, Inventiones Math.147 (2002), 671–691. Erratum 163 (2006), 227–228.

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Rade T. Živaljević.

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Živaljević, R.T. Oriented matroids and Ky Fan’s theorem. Combinatorica 30, 471–484 (2010). https://doi.org/10.1007/s00493-010-2446-x

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Mathematics Subject Classification (2000)

  • 52C40