Whiskers and sequentially Cohen-Macaulay graphs

Details Whiskers and sequentially Cohen-Macaulay graphs Whiskers and sequentially Cohen-Macaulay graphs

2006-05-17

arxiv.org/abs/math/0605487v2

J. Combin. Theory Ser. A 115 (2008), no. 2, 304-316

Mathematics

Commutative Algebra

final version: contains minor revisions, improved exposition

Scientific paper

Let G be a simple (i.e., no loops and no multiple edges) graph. We investigate the question of how to modify G combinatorially to obtain a sequentially Cohen-Macaulay graph. We focus on modifications given by 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 in G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G \cup 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 \cup W(S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if G \backslash S is not sequentially Cohen-Macaulay, then G \cup 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.

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