Archiv der Mathematik

, 93:451

Sequentially Cohen–Macaulay bipartite graphs: vertex decomposability and regularity

Article

Abstract

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen–Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo–Mumford regularity of R/I(G) can be determined from the invariants of G.

Mathematics Subject Classification (2000)

13F55 13D02 05C75 

Keywords

Sequentially Cohen–Macaulay Edge ideals Bipartite graphs Vertex decomposable Shellable complex Castelnuovo–Mumford regularity 

References

  1. 1.
    Björner A., Wachs M.: Shellable nonpure complexes and posets. I. Trans. Amer. Math. Soc. 348, 1299–1327 (1996)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Björner A., Wachs M.: Shellable nonpure complexes and posets. II. Trans. Amer. Math. Soc. 349, 3945–3975 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Carrá Ferro and D. Ferrarello, Cohen–Macaulay graphs arising from digraphs, (2007) arXiv:math/0703417v1Google Scholar
  4. 4.
    Dochtermann A., Engström A., Algebraic properties of edge ideals via combinatorial topology, Electron. J. Combin. 16 (2009), no. 2.Google Scholar
  5. 5.
    Estrada M., Villarreal R.H.: Cohen–Macaulay bipartite graphs. Arch. Math. 68, 124–128 (1997)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Faridi S.: Simplicial trees are sequentially Cohen–Macaulay. J. Pure Appl. Algebra 190, 121–136 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Francisco C.A., Hà H.T.: Whiskers and sequentially Cohen–Macaulay graphs. J. Combin. Theory Ser. A. 115, 304–316 (2008)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Francisco C.A., Hà H.T., Van Tuyl A.: Splittings of monomial ideals. Proc. Amer. Math. Soc. 137, 3271–3282 (2009)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Francisco C.A., Van Tuyl A.: Sequentially Cohen–Macaulay edge ideals. Proc. Amer. Math. Soc. 135, 2327–2337 (2007)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hà H.T., Van Tuyl A.: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. J. Algebraic Combin. 27, 215–245 (2008)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Herzog, T. Hibi, and H. Ohsugi, Unmixed bipartite graphs and sublattices of the Boolean lattices, To appear in J. Algebraic Combin. (2009) arXiv:0806.1088v1Google Scholar
  12. 12.
    Herzog J., Hibi T.: Distributive lattices, bipartite graphs and Alexander duality. J. Algebraic Combin. 22, 289–302 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Herzog J., Hibi T., Zheng X.: Dirac’s theorem on chordal graphs and Alexander duality. European J. Combin. 25, 949–960 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Katzmann M.: Characteristic-independence of Betti numbers of graph ideals. J. Combin. Theory Ser. A 113, 435–454 (2006)CrossRefMathSciNetGoogle Scholar
  15. 15.
    M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, To appear in J. Algebraic Combin. (2009). arXiv.09002.0437v1Google Scholar
  16. 16.
    F. Mohammadi and S. Moradi, Resolutions of unmixed bipartite graphs, arXiv:0901.3015v1Google Scholar
  17. 17.
    Provan J., Billera L.: Decompositions of simplicial complexes related to diameters of convex polyhedra. Math. Oper. Res. 5, 576–594 (1980)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R.P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progress in Mathematics 41. Birkhäuser Boston, Inc., Boston, MA, 1996.Google Scholar
  19. 19.
    N. Terai, Alexander duality theorem and Stanley–Reisner rings, Sürikaisekikenkyüsho Kökyüruko (1999), no. 1078, 174–184, Free resolutions of coordinate rings of projective varieties and related topics (Kyoto 1998).Google Scholar
  20. 20.
    Van Tuyl A., Villarreal R.H.: Shellable graphs and sequentially Cohen–Macaulay bipartite graphs. J. Combin. Theory Ser. A 115, 799–814 (2008)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Villarreal R.H.: Cohen–Macaulay graphs. Manuscripta Math. 66, 277–293 (1990)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    R. H. Villarreal, Monomial Algebras. Marcel Dekker, 2001Google Scholar
  23. 23.
    Villarreal R.H.: Unmixed bipartite graphs. Rev. Colombiana Mat. 41, 393–395 (2007)MATHMathSciNetGoogle Scholar
  24. 24.
    Woodroofe R.: Vertex decomposable graphs and obstructions to shellability. Proc. Amer. Math. Soc. 137, 3235–3246 (2009)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Zheng X.: Resolutions of Facet Ideals. Comm. Algebra 32, 2301–2324 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesLakehead UniversityThunder BayCanada

Personalised recommendations