Partial coloring, vertex decomposability and sequentially Cohen-Macaulay simplicial complexes. (English) Zbl 1328.05207

Summary: 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.


05E45 Combinatorial aspects of simplicial complexes
05A15 Exact enumeration problems, generating functions
13C14 Cohen-Macaulay modules
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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