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Sequentially Cohen-Macaulay edge ideals

Authors: Christopher A. Francisco and Adam Van Tuyl
Journal: Proc. Amer. Math. Soc. 135 (2007), 2327-2337
MSC (2000): Primary 13F55, 13D02, 05C38, 05C75
Published electronically: March 21, 2007
MathSciNet review: 2302553
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Abstract: Let $ G$ be a simple undirected graph on $ n$ vertices, and let $ \mathcal I(G) \subseteq R = k[x_1,\ldots,x_n]$ denote its associated edge ideal. We show that all chordal graphs $ G$ are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of $ \mathcal I(G)$ is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

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Additional Information

Christopher A. Francisco
Affiliation: Department of Mathematics, Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65203

Adam Van Tuyl
Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1, Canada

Keywords: Componentwise linear, sequentially Cohen-Macaulay, edge ideals, chordal graphs
Received by editor(s): November 1, 2005
Received by editor(s) in revised form: April 6, 2006
Published electronically: March 21, 2007
Communicated by: Michael Stillman
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.