Shellability of chessboard complexes

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 Bjorner & Wachs.