An Efficient Algebraic Criterion for Shellability

TL;DR

This paper introduces a new algebraic criterion for determining shellability of simplicial complexes, characterizes leaves algebraically, and shows that the face ring of Gallai simplicial complexes of trees is Cohen-Macaulay.

Contribution

It provides an efficient algebraic method for shellability and characterizes leaves, also introducing Gallai-simplicial complexes and analyzing their algebraic properties.

Findings

New algebraic criterion for shellability

Algebraic characterization of leaves in simplicial complexes

Gallai-simplicial complex of trees has Cohen-Macaulay face ring

Abstract

In this paper, we give a new and efficient algebraic criterion for the pure as well as non-pure shellability of simplicial complex $\Delta$ over [n]. We also give an algebraic characterization of a leaf in a simplicial complex (defined in [8]). Moreover, we introduce the concept of Gallai-simplicial complex $\Delta_{\Gamma}(G)$ of a finite simple graph G. As an application, we show that the face ring of the Gallai simplicial complex associated to tree is Cohen-Macaulay.