$$f$$ -Vectors Implying Vertex Decomposability

Abstract

We prove that if a pure simplicial complex of dimension d with n facets has the least possible number of (d − 1)-dimensional faces among all complexes with n faces of dimension d, then it is vertex decomposable. This answers a question of J. Herzog and T. Hibi. In fact, we prove a generalization of their theorem using combinatorial methods.