Whiskers and sequentially Cohen-Macaulay graphs |
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Authors: | Christopher A. Francisco,Huy Tà i Hà |
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Affiliation: | a Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA b Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, USA |
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Abstract: | 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∪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∪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∪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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Keywords: | Edge ideals of graphs Alexander duality Sequential Cohen-Macaulayness |
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