Balanced vertex decomposable simplicial complexes and their h-vectors

Abstract

Given any finite simplicial complex \Delta, we show how to construct a new simplicial complex \Delta_{\chi} that is balanced and vertex decomposable. Moreover, we show that the h-vector of the simplicial complex \Delta_{\chi} is precisely the f-vector, denoted f(\Delta), of the original complex \Delta. We deduce this result by relating f(\Delta) with the graded Betti numbers of the Alexander dual of \Delta_{\chi}. Our construction generalizes the "whiskering" construction of Villarreal, and Cook and Nagel. As a corollary of our work, we add a new equivalent statement to a theorem of Bj\"orner, Frankl, and Stanley that classifies the f-vectors of simplicial complexes. We also prove a special case of a conjecture of Cook and Nagel, and Constantinescu and Varbaro on the h-vectors of flag complexes.