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

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 for building 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.