Sequentially simplicial complexes and sequentially 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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10646-9

Published electronically:
November 18, 2010

MathSciNet review:
2775376

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Abstract: We introduce sequentially modules over a commutative graded ring and sequentially simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition . In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially if and only if its pure -skeleton is for all . For , we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first 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 cycles showing that the only sequentially cycles are odd cycles and, for , no cycle is sequentially with the exception of cycles of length and . We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially . We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially . 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:
https://doi.org/10.1090/S0002-9939-2010-10646-9

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.