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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- Christopher A. Francisco
- Affiliation: Department of Mathematics, Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65203
- MR Author ID: 719806
- Email: firstname.lastname@example.org
- Adam Van Tuyl
- Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1, Canada
- MR Author ID: 649491
- ORCID: 0000-0002-6799-6653
- Email: email@example.com
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Proc. Amer. Math. Soc. 135 (2007), 2327-2337
- MSC (2000): Primary 13F55, 13D02, 05C38, 05C75
- DOI: https://doi.org/10.1090/S0002-9939-07-08841-7
- MathSciNet review: 2302553