Israel Journal of Mathematics

, Volume 87, Issue 1–3, pp 97–110

Shellability of chessboard complexes

  • Günter M. Ziegler


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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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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