Advertisement

Combinatorica

, Volume 28, Issue 3, pp 315–323 | Cite as

Kruskal-Katona type theorems for clique complexes arising from chordal and strongly chordal graphs

  • Jürgen HerzogEmail author
  • Satoshi Murai
  • Xinxian Zheng
  • Takayuki Hibi
  • Ngô Viêt Trung
Article

Abstract

A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal-Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented.

Mathematics Subject Classification (2000)

05D05 05C69 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W. Bruns and J. Herzog: Cohen-Macaulay rings, Revised Edition, Cambridge University Press, 1996.Google Scholar
  2. [2]
    A. Björner: The unimodality conjecture for convex polytopes, Bull. Amer. Math. Soc. 4 (1981), 187–188.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    R. Charney and M. Davis: The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold; Pacific J. Math. 171 (1995), 117–137.zbMATHMathSciNetGoogle Scholar
  4. [4]
    J. Eckhoff: Über kombinatorisch-geometrische Eigenschaften von Komplexen und Familien konvexer Mengen, J. Reine Angew. Math. 313 (1980), 171–188.zbMATHMathSciNetGoogle Scholar
  5. [5]
    S. Faridi: Cohen-Macaulay properties of square-free monomial ideals, J. Combin. Theory, Ser. A 109 (2005), 299–329.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    D. Ferrarello and R. Fröberg: The Hilbert series of the clique complex, Graphs Combin. 21 (2005), 401–405.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Frohmader: Face vectors of flag complexes, arXiv:math/0605673, preprint.Google Scholar
  8. [8]
    S. R. Gal: Real root conjecture fails for five-and higher-dimensional spheres, Discrete Comput. Geom. 34 (2005), 269–284.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Herzog, T. Hibi and X. Zheng: Dirac’s theorem on chordal graphs and Alexander duality, European J. Combin. 25 (2004), 949–960.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    T. Hibi: What can be said about pure O-sequences?, J. Combin. Theory, Ser. A 50 (1989), 319–322.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    T. Hibi: Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, Glebe, N.S.W., Australia, 1992.zbMATHGoogle Scholar
  12. [12]
    G. Kalai: Characterization of f-vectors of families of convex sets in ℝd, Part I: Necessity of Eckhoff’s conditions; Israel J. Math. 48 (1984), 175–195.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    G. Kalai: Characterization of f-vectors of families of convex sets in ℝd, Part II: Sufficiency of Eckhoff’s conditions; J. Combin. Theory, Ser. A 41 (1986), 167–188.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    T. A. McKee and F. R. McMorris: Topics in Intersection Graph Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.zbMATHGoogle Scholar
  15. [15]
    P. Renteln: The Hilbert series of the face ring of a flag complex, Graphs Combin. 18 (2002), 605–619.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    R. Stanley: Combinatorics and Commutative Algebra, Second Edition, Birkhäuser, 1995.Google Scholar
  17. [17]
    X. Zheng: Resolutions of Facet Ideals, Comm. Algebra 32 (2004), 2301–2324.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Jürgen Herzog
    • 1
    Email author
  • Satoshi Murai
    • 2
  • Xinxian Zheng
    • 1
  • Takayuki Hibi
    • 2
  • Ngô Viêt Trung
    • 3
  1. 1.Fachbereich Mathematik und InformatikUniversität Duisburg-EssenEssenGermany
  2. 2.Department of Pure and Applied Mathematics Graduate School of Information Science and TechnologyOsaka UniversityToyonaka, OsakaJapan
  3. 3.Institute of MathematicsHanoiVietnam

Personalised recommendations