Open Access Open Access  Restricted Access Subscription Access

$k$-shellable simplicial complexes and graphs

Rahim Rahmati-Asghar

Abstract


In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying an expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture.

Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.


Full Text:

PDF

References


Bayati, S. and Herzog, J., Expansions of monomial ideals and multigraded modules, Rocky Mountain J. Math. 44 (2014), no. 6, 1781–1804. https://doi.org/10.1216/RMJ-2014-44-6-1781

Björner, A. and Wachs, M. L., Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327. https://doi.org/10.1090/S0002-9947-96-01534-6

Bruggesser, H. and Mani, P., Shellable decompositions of cells and spheres, Math. Scand. 29 (1971), 197–205. https://doi.org/10.7146/math.scand.a-11045

Castrillón, I. D. and Cruz, R., Escalonabilidad de grafos e hipergrafos simples que contienen vétices simpliciales, Matemáticas XX (2012), no. 1, 69–80.

Conca, A. and Herzog, J., Castelnuovo-Mumford regularity of products of ideals, Collect. Math. 54 (2003), no. 2, 137–152.

Cruz, R. and Estrada, M., Vértices simpliciales y escalonabilidad de grafos, Morfismos 12 (2008), no. 2, 17–32.

Dirac, G. A., On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71–76. https://doi.org/10.1007/BF02992776

Dress, A., A new algebraic criterion for shellability, Beiträge Algebra Geom. 34 (1993), no. 1, 45–55.

Duval, A. M., Goeckner, B., Klivans, C. J., and Martin, J. L., A non-partitionable Cohen-Macaulay simplicial complex, Adv. Math. 299 (2016), 381–395. https://doi.org/10.1016/j.aim.2016.05.011

Emtander, E., Mohammadi, F., and Moradi, S., Some algebraic properties of hypergraphs, Czechoslovak Math. J. 61(136) (2011), no. 3, 577–607. https://doi.org/10.1007/s10587-011-0031-0

Herzog, J. and Popescu, D., Finite filtrations of modules and shellable multicomplexes, Manuscripta Math. 121 (2006), no. 3, 385–410. https://doi.org/10.1007/s00229-006-0044-4

Herzog, J. and Takayama, Y., Resolutions by mapping cones, Homology Homotopy Appl. 4 (2002), no. 2, The Roos Festschrift volume, part 2, 277–294.

Lekkerkerker, C. G. and Boland, J. C., Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51 (1962/1963), 45–64. https://doi.org/10.4064/fm-51-1-45-64

Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986.

Rahmati-Asghar, R., Cohen-Macaulay simplicial complexes of degree $k$, Bull. Malays. Math. Sci. Soc. (2) 37 (2014), no. 1, 93–102.

Rahmati-Asghar, R., Cohen-Macaulay-ness in codimension for simplicial complexes and expansion functor, Bull. Iranian Math. Soc. 42 (2016), no. 1, 223–232.

Schrijver, A., Combinatorial optimization. Polyhedra and efficiency. Vol. A, Algorithms and Combinatorics, vol. 24, Springer-Verlag, Berlin, 2003.

Stanley, R. P., Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193. https://doi.org/10.1007/BF01394054

Stanley, R. P., Combinatorics and commutative algebra, second ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996.

Van Tuyl, A. and Villarreal, R. H., Shellable graphs and sequentially Cohen-Macaulay bipartite graphs, J. Combin. Theory Ser. A 115 (2008), no. 5, 799–814. https://doi.org/10.1016/j.jcta.2007.11.001




DOI: http://dx.doi.org/10.7146/math.scand.a-102975

Refbacks

  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.
OK


ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library