f-vectors implying vertex decomposability

TL;DR

This paper proves that pure simplicial complexes with minimal (d-1)-faces are vertex decomposable, answering a question by Herzog and Hibi, and generalizes their theorem using combinatorial methods.

Contribution

It establishes a new characterization of minimal face complexes as vertex decomposable and extends previous results through a broader combinatorial framework.

Findings

Pure complexes with minimal (d-1)-faces are vertex decomposable.

The result answers a question posed by Herzog and Hibi.

The proof uses combinatorial methods to generalize earlier theorems.

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.