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Cohen-macaulay graphs

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Abstract

For a graph G we consider its associated ideal I(G). We uncover large classes of Cohen-Macaulay (=CM) graphs, in particular the full subclass of CM trees is presented. A formula for the Krull dimension of the symmetric algebra of I(G) is given along with a description of when this algebra is a domain. The first Koszul homology module of a CM tree is also studied.

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References

  1. N. Deo,Graph Theory with Applications to Engineering and Computer Science, Prentice Hall series in Automatic Computation, 1974

  2. J. A. Eagon and M. Hochster, R-sequences and indeterminates, Quart. J. Math. Oxford25 (1974), 61–71

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Fröberg, A study of graded extremal rings and of monomial rings, Math. Scand.51 (1982), 22–34

    MATH  MathSciNet  Google Scholar 

  4. F. Harary,Graph Theory, Addison-Wesley, Reading, MA, 1972

    Book  Google Scholar 

  5. J. Herzog, A. Simis and W. V. Vasconcelos, Koszul homology and blowing up rings, Proc. Trento Commutative Algebra Conf., Lectures Notes in Pure and Applied Math., vol.84, Dekker, New York, 1983, 79–169

    Google Scholar 

  6. J. Herzog, A. Simis and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc.283 (1984), 661–683

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Huneke and M. E. Rossi, The dimension and components of symmetric algebras, J. Algebra98 (1986), 200–210

    Article  MATH  MathSciNet  Google Scholar 

  8. I. Kaplansky,Commutative Rings, University of Chicago Press, Chicago, 1974

    MATH  Google Scholar 

  9. H. Matsumura,Commutative Algebra, Benjamin/Cummings, Reading, MA, 1980

    MATH  Google Scholar 

  10. L. Mirsky and H. Perfect, Systems of representatives, J. Math. Anal. Applic.15 (1966), 520–568

    Article  MATH  MathSciNet  Google Scholar 

  11. M. D. Plummer, On a family of line-critical graphs. Monatsh. Math.71 (1967), 40–48

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Relsner, Cohen-Macaulay quotients of polynomial rings, Adv. Math.21 (1976), 31–49

    Google Scholar 

  13. A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math.103 (1981), 203–224

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Simis and W. V. Vasconcelos, Krull dimension and integrality of symmetric algebras, Manuscripta Math.61 (1988), 63–75

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Stanley,Combinatorics and Commutative Algebra, Birkhäuser, Boston, 1983

    MATH  Google Scholar 

  16. D. Taylor, Ideals generated by monomials in an R-sequence, Thesis, University of Chicago, 1966

  17. W. V. Vasconcelos, Koszul homology and the structure of low codimension Cohen-Macaulay ideals, Trans. Amer. Math. Soc.301 (1987), 591–613

    Article  MATH  MathSciNet  Google Scholar 

  18. W. V. Vasconcelos, Symmetric Algebras, Preprint, 1988

Download references

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Authors and Affiliations

  1. Departamento de Matemáticas Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Unidad Adolfo López Mateos, 07300, México, D.F.

    Rafael H. Villarreal

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  1. Rafael H. Villarreal
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Partially supported by SNI-SEP, COFAA-IPN and CONACyT.

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Villarreal, R.H. Cohen-macaulay graphs. Manuscripta Math 66, 277–293 (1990). https://doi.org/10.1007/BF02568497

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  • DOI: https://doi.org/10.1007/BF02568497

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