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Collectanea Mathematica

, Volume 70, Issue 2, pp 187–195 | Cite as

Regularity of symbolic powers of cover ideals of graphs

  • S. A. Seyed FakhariEmail author
Article
  • 34 Downloads

Abstract

Let G be a graph which belongs to either of the following classes: (i) bipartite graphs, (ii) unmixed graphs, or (iii) claw–free graphs. Assume that J(G) is the cover ideal G and \(J(G)^{(k)}\) is its k-th symbolic power. We prove that
$$\begin{aligned} k\mathrm{deg}(J(G))\le \mathrm{reg}(J(G)^{(k)})\le (k-1)\mathrm{deg}(J(G))+|V(G)|-1. \end{aligned}$$
We also determine families of graphs for which the above inequalities are equality.

Keywords

Cover ideal Regularity Symbolic power 

Mathematics Subject Classification

Primary 13D02 05E99 

Notes

Acknowledgements

The author thanks the referee for careful reading of the paper and for useful comments.

References

  1. 1.
    Alilooee, A., Banerjee, A.: Powers of edge ideals of regularity tree bipartite graphs. J. Commut. Algebra 9, 441–454 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alilooee, A., Beyarslan, S., Selvaraja, S.: Regularity of powers of unicyclic graphs (preprint)Google Scholar
  3. 3.
    Banerjee, A.: The regularity of powers of edge ideals. J. Algebr. Comb. 41, 303–321 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beyarslan, S., Hà, H.T., Trung, T.N.: Regularity of powers of forests and cycles. J. Algebr. Comb. 42, 1077–1095 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Conca, A.: Regularity jumps for powers of ideals. In: Commutative algebra, 2132, Lecture Notes Pure and Applied Mathematics, Queens Papers in Pure and Applied Mathematics, vol. 244, pp. 1–40. Chapman Hall/CRC, Boca Raton, FL, 2006. Math. 20, Kingston, Ontario: Queens University (1969)Google Scholar
  6. 6.
    Crupi, M., Rinaldo, G., Terai, N.: Cohen–Macaulay edge ideals whose height is half of the number of vertices. Nagoya Math. J. 201, 117–131 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cutkosky, D., Herzog, J., Trung, N.V.: Asymptotic behaviour of Castelnuovo–Mumford regularity. Compos. Math. 118, 243–261 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebr. Comb. 38, 37–55 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gitler, I., Reyes, E., Villarreal, R.H.: Blowup algebras of ideals of vertex covers of bipartite graphs. Contemp. Math. 376, 273–279 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gitler, I., Valencia, C.E.: Bounds for invariants of edge-rings. Commun. Algebra 33(5), 1603–1616 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hà, H.T., Trung, N.V., Trung, T.N.: Depth and regularity of powers of sums of ideals. Math. Z. 282, 819–838 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hang, N.T., Trung, T.N.: Regularity of powers of cover ideals of unimodular hypergraphs. J. Algebra 513, 159–176 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Herzog, J., Hibi, T.: Monomial Ideals. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    Jayanthan, A.V., Narayanan, N., Selvaraja, S.: Regularity of powers of bipartite graphs. J. Algebraic Combin. 47, 17–38 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moghimian, M., Seyed Fakhari, S.A., Yassemi, S.: Regularity of powers of edge ideal of whiskered cycles. Commun. Algebra 45, 1246–1259 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Seyed Fakhari, S.A.: Depth and Stanley depth of symbolic powers of cover ideals of graphs. J. Algebra 492, 402–413 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Seyed Fakhari, S.A.: Depth, Stanley depth and regularity of ideals associated to graphs. Arch. Math. (Basel) 107, 461–471 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Seyed Fakhari, S.A.: Symbolic powers of cover ideal of very well-covered and bipartite graphs. Proc. Am. Math. Soc. 146, 97–110 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2018

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran

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