Whiskers and sequentially Cohen-Macaulay graphs

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:115

Article

Francisco, Christopher A.¹  Ha, Huy Tai² 

[1] Oklahoma State Univ, Dept Math, Stillwater, OK 74078 USA

[2] Tulane Univ, Dept Math, New Orleans, LA 70118 USA

关键词: edge ideals of graphs;    Alexander duality;    sequential Cohen-Macaulayness;   

DOI  :  10.1016/j.jcta.2007.06.004

来源: Elsevier

【 摘 要 】

We investigate how to modify a simple graph G combinatorially to obtain a sequentially Cohen-Macaulay graph. We focus on adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex of G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G boolean OR W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen-Macaulay graph. For instance, we show that if S is a vertex cover of G, then G boolean OR W(S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if G \ S is not sequentially Cohen-Macaulay, then G boolean OR W(S) is not a sequentially Cohen-Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen-Macaulay graphs. (c) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free