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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
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 $ 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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  • 1. A. Björner and M. Wachs, Shellable nonpure complexes and posets, I, Trans. Amer. Math. Soc. 348 (1996), 1299-1327. MR 1333388 (96i:06008)
  • 2. A. Björner and M. Wachs, Shellable nonpure complexes and posets, II, Trans. Amer. Math. Soc. 349 (1997), 3945-3975. MR 1401765 (98b:06008)
  • 3. A. Björner, M. Wachs and V. Welker, On sequentially Cohen-Macaulay complexes and posets, Israel J. Math. 169 (2009), 295-316. MR 2460907 (2009m:05197)
  • 4. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1998. MR 1251956 (95h:13020)
  • 5. A. M. Duval, Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes, Electronic J. Combin. 3 (1996), 1-13. MR 1399398 (98b:06009)
  • 6. J. Eagon and V. Reiner, Resolution of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), 265-275. MR 1633767 (99h:13017)
  • 7. C. A. Francisco and H. T. Hà, Whiskers and sequentially Cohen-Macaulay graphs, J. Combin. Theory Ser. A 115 (2008), 304-316. MR 2382518 (2008j:13050)
  • 8. C. A. Francisco and A. Van Tuyl, Sequentially Cohen-Macaulay edge ideals, Proc. Amer. Math. Soc. 135 (2007), 2327-2337. MR 2302553 (2008a:13030)
  • 9. H. Haghighi, S. Yassemi and R. Zaare-Nahandi, Bipartite $ S_2$ graphs are Cohen-Macaulay, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53 (2010), 125-132.
  • 10. J. Herzog and T. Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141-153. MR 1684555 (2000i:13019)
  • 11. J. Herzog, T. Hibi and X. Zheng, Cohen-Macaulay chordal graphs, J. Combin. Theory Ser. A 113 (2006), 911-916. MR 2231097 (2007b:13042)
  • 12. J. Munkres, Topological results in combinatorics, Michigan Math. J. 31 (1994), 113-128. MR 736476 (85k:13022)
  • 13. J. S. Provan and L. J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), 576-594. MR 593648 (82c:52010)
  • 14. H. Sabzrou, M. Tousi and S. Yassemi, Simplicial join via tensor products, Manuscripta Math. 126 (2008), 255-272. MR 2403189 (2009c:13021)
  • 15. P. Schenzel, Dualisierende Komplexe in der lokalen Algbera und Buchsbaum-Ringe, LNM 907, Springer, 1982. MR 654151 (83i:13013)
  • 16. R. Stanley, Combinatorics and Commutative Algebra, Second Edition, Birkhäuser, Boston, 1995. MR 1453579 (98h:05001)
  • 17. N. Terai, Alexander duality in Stanley-Reisner rings, in Affine Algebraic Geometry (T. Hibi, ed.), Osaka University Press, Osaka, 2007, 449-462. MR 2330484 (2008d:13033)
  • 18. M. Tousi and S. Yassemi, Tensor products of some special rings, J. Algebra 268 (2003), 672-676. MR 2009326 (2005a:13047)
  • 19. A. Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity, Arch. Math. (Basel) 93 (2009), 451-459. MR 2563591
  • 20. A. Van Tuyl and R. H. Villarreal, Shellable graphs and sequentially Cohen-Macaulay bipartite graphs, J. Combin. Theory Ser. A 115 (2008), 799-814. MR 2417022 (2009b:13056)
  • 21. R. Woodroofe, Vertex decomposable graphs and obstructions to shellability, Proc. Amer. Math. Soc. 137 (2009), 3235-3246. MR 2515394 (2010e:05324)
  • 22. K. Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree $ \mathbb{N}^n$-graded modules, J. Algebra 225 (2000), 630-645. MR 1741555 (2000m:13036)
  • 23. K. Yanagawa, Dualizing complex of the face ring of a simplicial poset, arXiv:0910.1498

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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.

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