Israel Journal of Mathematics

, Volume 87, Issue 1–3, pp 97–110

Shellability of chessboard complexes

  • Günter M. Ziegler
Article

Abstract

The matchings in a complete bipartite graph form a simplicial complex, which in many cases has strong structural properties. We use an equivalent description aschessboard complexes: the complexes of all nontaking rook positions on chessboards of various shapes.

In this paper we construct ‘certificatek-shapes’ Σ(m, n, k) such that if the shapeA contains some Σ(m, n, k), then the (k−1)-skeleton of the chess-board complexδ(A) isvertex decomposable in the sense of Provan & Billera. This covers, in particular, the case of rectangular chessboardsA=[m]×[n], for which Δ(A) is vertex decomposable ifn≥2m−1, and the\(([\frac{{m + n + 1}}{3}] - 1)\)-skeleton is vertex decomposable in general.

The notion of vertex decomposability is a very convenient tool to prove shellability of such combinatorially defined simplicial complexes. We establish a relation between vertex decomposability and the CL-shellability technique (for posets) of Björner & Wachs.

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References

  1. [BP]
    L. J. Billera and J. S. Provan,A decomposition property for simplicial complexes and its relation to diameters and shellings, Ann. NY Acad. Sci.319 (1979), 82–85.CrossRefMathSciNetGoogle Scholar
  2. [Bj1]
    A. Björner,Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Advances in Math.52 (1984), 173–212.MATHCrossRefGoogle Scholar
  3. [Bj2]
    A. Björner,Homology and shellability of matroids and geometric lattices, inMatroid Applications (N. White, ed.), Cambridge University Press, 1992, pp. 226–283.Google Scholar
  4. [Bj3]
    A. Björner,topological Methods, inHandbook of Combinatorics (R. Graham, M. Grötschel and L. Lovász, eds.), North-Holland, Amsterdam, to appear.Google Scholar
  5. [Bj4]
    A. Björner, personal communication.Google Scholar
  6. [BLVZ]
    A. Björner, L. Lovász, S. T. Vrećica and R. T. Živaljević,Chessboard complexes and matching complexes, J. London Math. Soc., to appear.Google Scholar
  7. [BjW]
    A. Björner and M. Wachs,On lexicographically shellable posets, Trans. Amer. Math. Soc.277 (1983), 323–341.MATHCrossRefMathSciNetGoogle Scholar
  8. [Ga]
    P. F. Garst,Cohen-Macaulay Complexes and Group Actions, Ph.D. Thesis, University of Wisconsin-Madison, 1979, 130 pp.Google Scholar
  9. [KK]
    V. Klee and P. Kleinschmidt,The d-step conjecture and its relatives, Math. Operations Research12 (1987), 718–755.MATHMathSciNetCrossRefGoogle Scholar
  10. [LP]
    L. Lovász and M. D. Plummer,Matching Theory, Akadémiai Kiadó, Budapest, and North-Holland, Amsterdam, 1986.MATHGoogle Scholar
  11. [PB]
    J. S. Provan and L. J. Billera,Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Operations Research5 (1980), 576–594.MATHMathSciNetGoogle Scholar
  12. [Sa]
    K. S. Sarkaria,A generalized van Kampen-Flores theorem, Proc. Amer. Math. Soc.111 (1991), 559–565.MATHCrossRefMathSciNetGoogle Scholar
  13. [VZ]
    S. T. Vrećica and R. T. Živaljević,The colored Tverberg’s problem and complexes of injective functions, J. Combinatorial Theory, Ser. A61 (1992), 309–318.CrossRefMATHGoogle Scholar
  14. [WW]
    M. L. Wachs and J. W. Walker,On geometric semilattices, Order2 (1986), 367–385.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1994

Authors and Affiliations

  • Günter M. Ziegler
    • 1
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)BerlinGermany

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