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SC 92-19

Günter M. Ziegler Shellability of Chessboard Complexes. Appeared in: Israel J. Mathematics 87 (1994) pp. 97-110

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 as chessboard complexes: the complexes of all non-taking rook positions on chessboards of various shapes.
In this paper we construct `certificate $k$-shapes $\Sigma(m,n,k)$ such that if the shape $A$ contains some $\Sigma(m,n,k)$, then the $(k{-}1)$-skeleton of the chessboard complex $\Delta(A)$ is vertex decomposable in the sense of Provan & Billera. This covers, in particular, the case of rectangular chessboards $A=[m]{\times}[n]$, for which $\Delta(A)$ is vertex decomposable if $n\ge 2m{-}1$, and the $(\lfloor{m+n+1\over3}\rfloor{-}1)$-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.