Abstract
Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rm sdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rm depth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).
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A. M. Duval, B. Goeckner, C. J. Klivans, and J. L. Martin, A non-partitionable Cohen-Macaulay simplicial complex, preprint.
Brodmann M.: The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc. 86, 35–39 (1979)
Burch L.: Codimension and analytic spread, Math. Proc. Cambridge Philos. Soc. 72, 369–373 (1972)
Constantinescu A., Pournaki M.R., Seyed Fakhari S.A., Terai N., Yassemi S.: Cohen-Macaulayness and limit behavior of depth for powers of cover ideals, Comm. Algebra 43, 143–157 (2015)
Constantinescu A., Varbaro M.: Koszulness, Krull dimension, and other properties of graph-related algebras, J. Algebraic Combin. 34, 375–400 (2011)
Cimpoeaş M.: Several inequalities regarding Stanley depth. Rom. J. Math. Comput. Sci. 2, 28–40 (2012)
Cimpoeaş M.: Stanley depth of monomial ideals with small number of generators, Cent. Eur. J. Math. 7, 629–634 (2009)
Dao H., Huneke C., Schweig J.: Bounds on the regularity, and projective dimension of ideals associated to graphs, J. Algebraic Combin. 38, 37–55 (2013)
Gitler I., Reyes E., Villarreal R.H.: Blowup algebras of ideals of vertex covers of bipartite graphs, Contemp. Math. 376, 273–279 (2005)
Hà H.T., Van Tuyl A.: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27, 215–245 (2008)
J. Herzog, A survey on Stanley depth, In: Monomial Ideals, Computations and Applications, A. Bigatti, P.Giménez, E. Sáenz-de-Cabezón (Eds.), Proceedings of MONICA 2011, Lecture Notes in Math. 2083, Springer, Heidelberg, 2013.
J. Herzog and T. Hibi, Monomial Ideals, Springer-Verlag, London, 2011.
Herzog J., Vladoiu M., Zheng X.: How to compute the Stanley depth of a monomial ideal, J. Algebra 322, 3151–3169 (2009)
Kummini M.: Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30, 429–445 (2009)
E. Nevo: Regularity of edge ideals of C 4-free graphs via the topology of the lcm-lattice, J. Combin. Theory Ser. A 118, 491–501 (2011)
I. Peeva, Graded syzygies, Algebra and Applications, vol. 14, Springer-Verlag London Ltd., London, 2011.
Popescu D.: Bounds of Stanley depth, An. Ştiinţ. Univ. ``Ovidius'' Constanţa Ser. Mat. 19, 187–194 (2011)
Pournaki M.R., Seyed Fakhari S.A., Tousi M., Yassemi S.: What is \({\ldots}\) Stanley depth? Notices Amer. Math. Soc. 56, 1106–1108 (2009)
Rauf A.: Stanley decompositions, pretty clean filtrations and reductions modulo regular elements, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 50(98), 347–354 (2007)
S. A. Seyed Fakhari, Stanley depth and symbolic powers of monomial ideals, Math. Scand., to appear.
Seyed Fakhari S.A.: Stanley depth of the integral closure of monomial ideals, Collect. Math. 64, 351–362 (2013)
Stanley R.P.: Linear Diophantine equations and local cohomology, Invent. Math. 68, 175–193 (1982)
Van Tuyl A.: Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity, Arch. Math. (Basel) 93, 451–459 (2009)
Woodroofe R.: Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commut. Algebra 6, 287–304 (2014)
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Seyed Fakhari, S.A. Depth, Stanley depth, and regularity of ideals associated to graphs. Arch. Math. 107, 461–471 (2016). https://doi.org/10.1007/s00013-016-0965-4
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DOI: https://doi.org/10.1007/s00013-016-0965-4