## Sequentially Cohen-Macaulay edge ideals

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- by Christopher A. Francisco and Adam Van Tuyl PDF
- Proc. Amer. Math. Soc.
**135**(2007), 2327-2337 Request permission

## 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.## References

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## Additional Information

**Christopher A. Francisco**- Affiliation: Department of Mathematics, Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65203
- MR Author ID: 719806
- Email: chrisf@math.missouri.edu
**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: avantuyl@sleet.lakeheadu.ca
- 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