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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Depth of edge rings arising from finite graphs


Authors: Takayuki Hibi, Akihiro Higashitani, Kyouko Kimura and Augustine B. O’Keefe
Journal: Proc. Amer. Math. Soc. 139 (2011), 3807-3813
MSC (2010): Primary 13P10
Published electronically: April 7, 2011
MathSciNet review: 2823027
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Abstract: Let $ G$ be a finite graph and $ K[G]$ the edge ring of $ G$. Based on the technique of Gröbner bases and initial ideals, it will be proved that, given integers $ f$ and $ d$ with $ 7 \leq f \leq d$, there exists a finite graph $ G$ on $ [d] = \{ 1, \ldots, d \}$ with $ \textrm{depth} K[G] = f$ and with $ \textrm{Krull-dim} K[G] = d$.


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Additional Information

Takayuki Hibi
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: hibi@math.sci.osaka-u.ac.jp

Akihiro Higashitani
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: sm5037ha@ecs.cmc.osaka-u.ac.jp

Kyouko Kimura
Affiliation: Department of Mathematics, Faculty of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan
Email: skkimur@ipc.shizuoka.ac.jp

Augustine B. O’Keefe
Affiliation: Department of Mathematics, Tulane University, 6823 St. Charles Avenue, New Orleans, Louisiana 70118
Email: aokeefe@tulane.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11083-9
Keywords: Edge ring, toric ideal, finite graph, Gröbner basis, initial ideal
Received by editor(s): September 8, 2010
Published electronically: April 7, 2011
Additional Notes: The fourth author had summer support provided by the JSPS Research Fellowships for Young Scientists and the NSF East Asia and Pacific Institutes Fellowship.
Communicated by: Irena Peeva
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.