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Acta Mathematica Vietnamica

, Volume 40, Issue 3, pp 511–526 | Cite as

Combinatorial Characterizations of the Saturation and the Associated Primes of the Fourth Power of Edge Ideals

  • Ha Thi Thu Hien
  • Ha Minh Lam
Article

Abstract

To compute the local cohomology of powers of edge ideals one needs to know their saturations. The saturation of the second and third powers has been described in terms of the graph in Hien et al. (J. Algebra 439, 225–244, 2015) and Terai and Trung (J. Pure Appl. Algebra 218, 1117–1129, 2014). In this article, we give a combinatorial description of the generators of the saturation of the fourth power. As a consequence, we are able to give a complete classification of the associated primes of the fourth power of edge ideals in terms of the graph.

Keywords

Edge ideal Power Associated prime Depth Dominating subgraph Odd cycle 

Mathematics Subject Classification (2010)

13C05 13C15 13F55 

Notes

Acknowledgments

The authors would like to thank Prof. Ngo Viet Trung for his guidance.

References

  1. 1.
    Berge, C.: Two theorems in graph theory. Proc. Natl. Acad. Sci. USA 43, 842–844 (1957)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, J., Morey, S., Sung, A.: The stable set of associated primes of the ideal of a graph. Rocky Mountain. J. Math. 32, 71–89 (2002)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Eisenbud, D.: Commutative Algebra: with a View Toward Algebraic Geometry. Springer-Verlag, New York (1995)Google Scholar
  4. 4.
    Francisco, C., Ha, H.T., Van Tuyl, A.: A conjecture on critical graphs and connections to the persistence of associated primes. Discret. Math. 310, 2176–2182 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Francisco, C., Ha, H.T., Van Tuyl, A.: Associated primes of monomial ideals and odd holes in graphs. J. Algebr. Comb. 32, 287–301 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Francisco, C., Ha, H.T., Van Tuyl, A.: Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals. J. Algebra 331, 224–242 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ha, H.T., Morey, S.: Embedded associated primes of powers of squarefree monomial ideals. J. Pure Appl. Algebra 214, 301–308 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ha, H.T., Sun, M.: Squarefree monomial ideals that fail the persistence property and non-increasing depth. Acta Math. Vietnam. 40(1), 125–137 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hien, H.T.T., Lam, H.M., Trung, N.V.: Saturation and associated primes of powers of edge ideals. J. Algebra 439, 225–244 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Martinez-Bernal, J., Morey, S., Villarreal, B.: Associated primes of powers of edge ideals. Collect. Math. 63, 361–374 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Takayama, Y.: Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48, 327–344 (2005)MathSciNetGoogle Scholar
  13. 13.
    Terai, N., Trung, N.V.: On the associated primes and the depth of the second power of squarefree monomial ideals. J. Pure Appl. Algebra 218, 1117–1129 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    West, D.B.: Introduction to Graph Theory, 2nd. Prentice-Hall (2001)Google Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Foreign Trade UniversityHanoiVietnam
  2. 2.Institute of MathematicsHanoiVietnam

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