Partial coloring, vertex decomposability, and sequentially Cohen-Macaulay simplicial complexes

TL;DR

This paper explores how coloring vertices in simplicial complexes affects their algebraic properties, providing conditions for vertex decomposability and applications to Cohen-Macaulay properties of graph ideals.

Contribution

It introduces a new construction based on vertex coloring to analyze vertex decomposability and sequential Cohen-Macaulayness of simplicial complexes.

Findings

Characterizes when the construction yields vertex decomposable complexes

Provides new proofs for Cohen-Macaulay properties of certain graph ideals

Strengthens understanding of algebraic properties via combinatorial modifications

Abstract

In attempting to understand how combinatorial modifications alter algebraic properties of monomial ideals, several authors have investigated the process of adding "whiskers" to graphs. In this paper, we study a similar construction to build a simplicial complex $\Delta_\chi$ from a coloring $\chi$ of a subset of the vertices of $\Delta$, and give necessary and sufficient conditions for this construction to produce vertex decomposable simplicial complexes. We apply this work to strengthen and give new proofs about sequentially Cohen-Macaulay edge ideals of graphs.