Sequentially simplicial complexes and sequentially graphs
Authors:
Hassan Haghighi, Naoki Terai, Siamak Yassemi and Rahim ZaareNahandi
Journal:
Proc. Amer. Math. Soc. 139 (2011), 19932005
MSC (2010):
Primary 13H10, 05C75
Published electronically:
November 18, 2010
MathSciNet review:
2775376
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
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 CohenMacaulay, and satisfying Serre's condition . In analogy with the sequentially CohenMacaulay 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 CohenMacaulay 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.
 1.
Anders
Björner and Michelle
L. Wachs, Shellable nonpure complexes and
posets. I, Trans. Amer. Math. Soc.
348 (1996), no. 4,
1299–1327. MR 1333388
(96i:06008), 10.1090/S0002994796015346
 2.
Anders
Björner and Michelle
L. Wachs, Shellable nonpure complexes and
posets. II, Trans. Amer. Math. Soc.
349 (1997), no. 10, 3945–3975. MR 1401765
(98b:06008), 10.1090/S0002994797018382
 3.
Anders
Björner, Michelle
Wachs, and Volkmar
Welker, On sequentially CohenMacaulay complexes and posets,
Israel J. Math. 169 (2009), 295–316. MR 2460907
(2009m:05197), 10.1007/s1185600900122
 4.
Winfried
Bruns and Jürgen
Herzog, CohenMacaulay rings, Cambridge Studies in Advanced
Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
(95h:13020)
 5.
Art
M. Duval, Algebraic shifting and sequentially CohenMacaulay
simplicial complexes, Electron. J. Combin. 3 (1996),
no. 1, Research Paper 21, approx.\ 14 pp.\ (electronic). MR 1399398
(98b:06009)
 6.
John
A. Eagon and Victor
Reiner, Resolutions of StanleyReisner rings and Alexander
duality, J. Pure Appl. Algebra 130 (1998),
no. 3, 265–275. MR 1633767
(99h:13017), 10.1016/S00224049(97)000972
 7.
Christopher
A. Francisco and Huy
Tài Hà, Whiskers and sequentially CohenMacaulay
graphs, J. Combin. Theory Ser. A 115 (2008),
no. 2, 304–316. MR 2382518
(2008j:13050), 10.1016/j.jcta.2007.06.004
 8.
Christopher
A. Francisco and Adam
Van Tuyl, Sequentially CohenMacaulay edge
ideals, Proc. Amer. Math. Soc.
135 (2007), no. 8,
2327–2337 (electronic). MR 2302553
(2008a:13030), 10.1090/S0002993907088417
 9.
H. Haghighi, S. Yassemi and R. ZaareNahandi, Bipartite graphs are CohenMacaulay, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53 (2010), 125132.
 10.
Jürgen
Herzog and Takayuki
Hibi, Componentwise linear ideals, Nagoya Math. J.
153 (1999), 141–153. MR 1684555
(2000i:13019)
 11.
Jürgen
Herzog, Takayuki
Hibi, and Xinxian
Zheng, CohenMacaulay chordal graphs, J. Combin. Theory Ser. A
113 (2006), no. 5, 911–916. MR 2231097
(2007b:13042), 10.1016/j.jcta.2005.08.007
 12.
James
R. Munkres, Topological results in combinatorics, Michigan
Math. J. 31 (1984), no. 1, 113–128. MR 736476
(85k:13022), 10.1307/mmj/1029002969
 13.
J.
Scott Provan and Louis
J. Billera, Decompositions of simplicial complexes related to
diameters of convex polyhedra, Math. Oper. Res. 5
(1980), no. 4, 576–594. MR 593648
(82c:52010), 10.1287/moor.5.4.576
 14.
Hossein
Sabzrou, Massoud
Tousi, and Siamak
Yassemi, Simplicial join via tensor product, Manuscripta Math.
126 (2008), no. 2, 255–272. MR 2403189
(2009c:13021), 10.1007/s002290080175x
 15.
Peter
Schenzel, Dualisierende Komplexe in der lokalen Algebra und
BuchsbaumRinge, Lecture Notes in Mathematics, vol. 907,
SpringerVerlag, BerlinNew York, 1982 (German). With an English summary.
MR 654151
(83i:13013)
 16.
Richard
P. Stanley, Combinatorics and commutative algebra, 2nd ed.,
Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc.,
Boston, MA, 1996. MR 1453579
(98h:05001)
 17.
Naoki
Terai, Alexander duality in StanleyReisner rings, Affine
algebraic geometry, Osaka Univ. Press, Osaka, 2007,
pp. 449–462. MR 2330484
(2008d:13033)
 18.
Masoud
Tousi and Siamak
Yassemi, Tensor products of some special rings, J. Algebra
268 (2003), no. 2, 672–676. MR 2009326
(2005a:13047), 10.1016/S00218693(03)001054
 19.
Adam
Van Tuyl, Sequentially CohenMacaulay bipartite graphs: vertex
decomposability and regularity, Arch. Math. (Basel)
93 (2009), no. 5, 451–459. MR 2563591
(2010j:13043), 10.1007/s0001300900499
 20.
Adam
Van Tuyl and Rafael
H. Villarreal, Shellable graphs and sequentially CohenMacaulay
bipartite graphs, J. Combin. Theory Ser. A 115
(2008), no. 5, 799–814. MR 2417022
(2009b:13056), 10.1016/j.jcta.2007.11.001
 21.
Russ
Woodroofe, Vertex decomposable graphs and
obstructions to shellability, Proc. Amer. Math.
Soc. 137 (2009), no. 10, 3235–3246. MR 2515394
(2010e:05324), 10.1090/S000299390909981X
 22.
Kohji
Yanagawa, Alexander duality for StanleyReisner rings and
squarefree 𝐍ⁿgraded modules, J. Algebra
225 (2000), no. 2, 630–645. MR 1741555
(2000m:13036), 10.1006/jabr.1999.8130
 23.
K. Yanagawa, Dualizing complex of the face ring of a simplicial poset, arXiv:0910.1498
 1.
 A. Björner and M. Wachs, Shellable nonpure complexes and posets, I, Trans. Amer. Math. Soc. 348 (1996), 12991327. MR 1333388 (96i:06008)
 2.
 A. Björner and M. Wachs, Shellable nonpure complexes and posets, II, Trans. Amer. Math. Soc. 349 (1997), 39453975. MR 1401765 (98b:06008)
 3.
 A. Björner, M. Wachs and V. Welker, On sequentially CohenMacaulay complexes and posets, Israel J. Math. 169 (2009), 295316. MR 2460907 (2009m:05197)
 4.
 W. Bruns and J. Herzog, CohenMacaulay Rings, Cambridge University Press, 1998. MR 1251956 (95h:13020)
 5.
 A. M. Duval, Algebraic shifting and sequentially CohenMacaulay simplicial complexes, Electronic J. Combin. 3 (1996), 113. MR 1399398 (98b:06009)
 6.
 J. Eagon and V. Reiner, Resolution of StanleyReisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), 265275. MR 1633767 (99h:13017)
 7.
 C. A. Francisco and H. T. Hà, Whiskers and sequentially CohenMacaulay graphs, J. Combin. Theory Ser. A 115 (2008), 304316. MR 2382518 (2008j:13050)
 8.
 C. A. Francisco and A. Van Tuyl, Sequentially CohenMacaulay edge ideals, Proc. Amer. Math. Soc. 135 (2007), 23272337. MR 2302553 (2008a:13030)
 9.
 H. Haghighi, S. Yassemi and R. ZaareNahandi, Bipartite graphs are CohenMacaulay, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53 (2010), 125132.
 10.
 J. Herzog and T. Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141153. MR 1684555 (2000i:13019)
 11.
 J. Herzog, T. Hibi and X. Zheng, CohenMacaulay chordal graphs, J. Combin. Theory Ser. A 113 (2006), 911916. MR 2231097 (2007b:13042)
 12.
 J. Munkres, Topological results in combinatorics, Michigan Math. J. 31 (1994), 113128. 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), 576594. MR 593648 (82c:52010)
 14.
 H. Sabzrou, M. Tousi and S. Yassemi, Simplicial join via tensor products, Manuscripta Math. 126 (2008), 255272. MR 2403189 (2009c:13021)
 15.
 P. Schenzel, Dualisierende Komplexe in der lokalen Algbera und BuchsbaumRinge, 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 StanleyReisner rings, in Affine Algebraic Geometry (T. Hibi, ed.), Osaka University Press, Osaka, 2007, 449462. MR 2330484 (2008d:13033)
 18.
 M. Tousi and S. Yassemi, Tensor products of some special rings, J. Algebra 268 (2003), 672676. MR 2009326 (2005a:13047)
 19.
 A. Van Tuyl, Sequentially CohenMacaulay bipartite graphs: vertex decomposability and regularity, Arch. Math. (Basel) 93 (2009), 451459. MR 2563591
 20.
 A. Van Tuyl and R. H. Villarreal, Shellable graphs and sequentially CohenMacaulay bipartite graphs, J. Combin. Theory Ser. A 115 (2008), 799814. MR 2417022 (2009b:13056)
 21.
 R. Woodroofe, Vertex decomposable graphs and obstructions to shellability, Proc. Amer. Math. Soc. 137 (2009), 32353246. MR 2515394 (2010e:05324)
 22.
 K. Yanagawa, Alexander duality for StanleyReisner rings and squarefree graded modules, J. Algebra 225 (2000), 630645. MR 1741555 (2000m:13036)
 23.
 K. Yanagawa, Dualizing complex of the face ring of a simplicial poset, arXiv:0910.1498
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
13H10,
05C75
Retrieve articles in all journals
with MSC (2010):
13H10,
05C75
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 8408502, Japan
Email:
terai@cc.sagau.ac.jp
Siamak Yassemi
Affiliation:
School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran
Email:
yassemi@ipm.ir
Rahim ZaareNahandi
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/S000299392010106469
Keywords:
Sequentially CohenMacaualy,
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.
