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

DOI:
https://doi.org/10.1090/S0002-9939-07-08841-7

Published electronically:
March 21, 2007

MathSciNet review:
2302553

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a simple undirected graph on vertices, and let denote its associated edge ideal. We show that all chordal graphs are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of 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

Email:
chrisf@math.missouri.edu

**Adam Van Tuyl**

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

Email:
avantuyl@sleet.lakeheadu.ca

DOI:
https://doi.org/10.1090/S0002-9939-07-08841-7

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.