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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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