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

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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.

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References

  1. W. Bruns and J. Herzog: Cohen-Macaulay rings, Revised Edition, Cambridge University Press, 1996.

  2. A. Björner: The unimodality conjecture for convex polytopes, Bull. Amer. Math. Soc. 4 (1981), 187–188.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Charney and M. Davis: The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold; Pacific J. Math. 171 (1995), 117–137.

    MATH  MathSciNet  Google Scholar 

  4. J. Eckhoff: Über kombinatorisch-geometrische Eigenschaften von Komplexen und Familien konvexer Mengen, J. Reine Angew. Math. 313 (1980), 171–188.

    MATH  MathSciNet  Google Scholar 

  5. S. Faridi: Cohen-Macaulay properties of square-free monomial ideals, J. Combin. Theory, Ser. A 109 (2005), 299–329.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Ferrarello and R. Fröberg: The Hilbert series of the clique complex, Graphs Combin. 21 (2005), 401–405.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Frohmader: Face vectors of flag complexes, arXiv:math/0605673, preprint.

  8. S. R. Gal: Real root conjecture fails for five-and higher-dimensional spheres, Discrete Comput. Geom. 34 (2005), 269–284.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Herzog, T. Hibi and X. Zheng: Dirac’s theorem on chordal graphs and Alexander duality, European J. Combin. 25 (2004), 949–960.

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Hibi: What can be said about pure O-sequences?, J. Combin. Theory, Ser. A 50 (1989), 319–322.

    Article  MATH  MathSciNet  Google Scholar 

  11. T. Hibi: Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, Glebe, N.S.W., Australia, 1992.

    MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. A. McKee and F. R. McMorris: Topics in Intersection Graph Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.

    MATH  Google Scholar 

  15. P. Renteln: The Hilbert series of the face ring of a flag complex, Graphs Combin. 18 (2002), 605–619.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Stanley: Combinatorics and Commutative Algebra, Second Edition, Birkhäuser, 1995.

  17. X. Zheng: Resolutions of Facet Ideals, Comm. Algebra 32 (2004), 2301–2324.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Universität Duisburg-Essen, Campus Essen, Essen, 45117, Germany

    Jürgen Herzog & Xinxian Zheng

  2. Department of Pure and Applied Mathematics Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Satoshi Murai & Takayuki Hibi

  3. Institute of Mathematics, Vien Toan Hoc 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    Ngô Viêt Trung

Authors
  1. Jürgen Herzog
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  2. Satoshi Murai
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  3. Xinxian Zheng
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  4. Takayuki Hibi
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  5. Ngô Viêt Trung
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Correspondence to Jürgen Herzog.

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Herzog, J., Murai, S., Zheng, X. et al. Kruskal-Katona type theorems for clique complexes arising from chordal and strongly chordal graphs. Combinatorica 28, 315–323 (2008). https://doi.org/10.1007/s00493-008-2319-8

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  • DOI: https://doi.org/10.1007/s00493-008-2319-8

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