Abstract
If a simplicial complex \(\Delta \) is vertex decomposable, then the n-sphere moment angle complex \(\mathcal {Z}_{\Delta ^{\vee }}(D^{n},S^{n-1})\) has the homotopy type of the wedges of spheres, where \(\Delta ^{\vee }\) is the Alexander dual of \(\Delta \). Furthermore, if \(\Delta \) is pure vertex decomposable, then its Stanley–Reisner ring \(k[\Delta ]\) is Cohen–Macaulay. Consequently, the vertex decomposable property is an interesting property from combinatorial, algebraic and topological point of view. In this paper, we characterize the pure vertex decomposable simplicial complexes associated to graphs whose 5-cycles have at least 4 chords.
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Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces. Adv. Math. 225, 1634–1668 (2010)
Bruns, A., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1998)
Buchstaber, V.M., Panov, T.E.: Torus actions and their applications in topology and combinatorics. In: University Lecture Series, vol. 24. Amer. Math. Soc., Providence (2002)
Castrillón, I.D., Cruz, R., Reyes, E.: On well-covered, vertex decomposable and Cohen–Macaulay. Electron. J. Comb. 23(2) (2016)
Crupi, M., Rinaldo, G., Terai, N.: Cohen–Macaulay edges ideals whose height is half of the number of vertices. Nagoya Math. J. 201, 116–130 (2011)
Davis, M.W., Januszkiewicz, T.: Convex polytopes, coxeter orbifolds and torus actions. Duke Math. J. 62, 417–452 (1991)
Earl, J., Vander Meulen, K.N., Van Tuyl, A.: Independence complexes of well-covered circulant graphs. Exp. Math. 25, 441–451 (2016)
Estrada, M., Villarreal, R.H.: Cohen–Macaulay bipartite graphs. Arch. Math. 68, 124–128 (1997)
Grujić, V., Welker, V.: Moment-angle complexes of pairs \((D^{n}, S^{n-1})\) and simplicial complexes with vertex decomposable duals. Monatsch. Math. 176(2), 255–273 (2015)
Herzog, J., Hibi, T.: Distributive lattices, bipartite graphs and Alexander duality. J. Alg. Combin. 22, 289–302 (2005)
Hoang, D.T., Minh, N.C., Trung, T.N.: Cohen–Macaulay graphs with large girth. J. Algebra Appl. 14(7), 1550112 (2015)
Iriye, K., Kishimoto, D.: Fat wedge filtrations and decomposition of polyhedral products. arXiv:1412.4866
Mahmoudi, M., Mousivand, A., Crupi, M., Rinaldo, G., Terai, N., Yassemi, S.: Vertex decomposability and regularity of very well-covered graphs. J. Pure Appl. Algebra 215(10), 2473–2480 (2011)
Morey, S., Reyes, E., Villarreal, R.H.: Cohen–Macaulay, shellable and unmixed clutters with perfect matching of König type. J. Pure Appl. Algebra 212(7), 1770–1786 (2008)
Randerath, B., Volkmann, L.: A characterization of well covered block-cactus graphs. Australas. J. Combin. 9, 307–314 (1994)
Stanley, R.P.:Combinatorics and Commutative Algebra. In: Progress in Mathematics, vol. 41, 2nd edn. Birkhäuser Boston, Inc., Boston (1996)
Villarreal, R.H.: Monomial Algebras. In: Monographs and Research Notes in Mathematics, 2nd edn. Chapman and Hall/CRC, New York (2015)
Woodroofe, R.: Vertex decomposable graphs and obstructions to shellability. Proc. Am. Math. Soc. 137(10), 3235–3246 (2009)
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The authors are grateful to the referees whose suggestions improved the presentation of this paper.
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In honor to Samuel Gitler.
I. D. Castrillón was supported by ABACUS-CINVESTAV.
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Castrillón, I.D., Reyes, E. Pure vertex decomposable simplicial complex associated to graphs whose 5-cycles are chorded. Bol. Soc. Mat. Mex. 23, 399–412 (2017). https://doi.org/10.1007/s40590-016-0147-1
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DOI: https://doi.org/10.1007/s40590-016-0147-1