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

 

Sequentially $ S_r$ simplicial complexes and sequentially $ S_2$ graphs


Authors: Hassan Haghighi, Naoki Terai, Siamak Yassemi and Rahim Zaare-Nahandi
Journal: Proc. Amer. Math. Soc. 139 (2011), 1993-2005
MSC (2010): Primary 13H10, 05C75
Published electronically: November 18, 2010
MathSciNet review: 2775376
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Abstract: We introduce sequentially $ S_r$ modules over a commutative graded ring and sequentially $ S_r$ simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition $ S_r$. In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially $ S_r$ if and only if its pure $ i$-skeleton is $ S_r$ for all $ i$. For $ r=2$, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially $ S_r$ if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first $ r$ steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially $ S_r$ cycles showing that the only sequentially $ S_2$ cycles are odd cycles and, for $ r\ge 3$, no cycle is sequentially $ S_r$ with the exception of cycles of length $ 3$ and $ 5$. We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially $ S_r$ graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially $ S_2$. We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially $ S_2$. Finally, we propose some questions.


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Additional Information

Hassan Haghighi
Affiliation: Department of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
Email: haghighi@kntu.ac.ir

Naoki Terai
Affiliation: Department of Mathematics, Faculty of Culture and Education, SAGA University, SAGA 840-8502, Japan
Email: terai@cc.saga-u.ac.jp

Siamak Yassemi
Affiliation: School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran
Email: yassemi@ipm.ir

Rahim Zaare-Nahandi
Affiliation: School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran
Email: rahimzn@ut.ac.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10646-9
PII: S 0002-9939(2010)10646-9
Keywords: Sequentially Cohen-Macaualy, Serre’s condition, sequentially $S_{r}$ simplicial complex
Received by editor(s): April 21, 2010
Received by editor(s) in revised form: June 7, 2010
Published electronically: November 18, 2010
Additional Notes: The first author was supported in part by a grant from K. N. Toosi University of Technology
The third author was supported in part by a grant from IPM (No. 89130214)
The fourth author was supported in part by a grant from the University of Tehran
Communicated by: Irena Peeva
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.