Ehrhart Series of Fractional Stable Set Polytopes of Finite Graphs

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Abstract

The fractional stable set polytope FRAC(G) of a simple graph G with d vertices is a rational polytope that is the set of nonnegative vectors (x1, . . . , x d ) satisfying x i xj \({\leq}\) 1 for every edge (i, j) of G. In this paper we show that (i) the \({\delta}\)-vector of a lattice polytope 2FRAC(G) is alternatingly increasing, (ii) the Ehrhart ring of FRAC(G) is Gorenstein, (iii) the coefficients of the numerator of the Ehrhart series of FRAC(G) are symmetric, unimodal and computed by the \({\delta}\)-vector of 2FRAC(G).

Mathematics Subject Classification

52B05 52B20 

Keywords

Ehrhart series Ehrhart rings fractional stable set polytopes Gorenstein Fano polytopes unimodal \({\delta}\)-vectors 

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References

  1. 1.
    Athanasiadis, C.A.: \({h^*}\)-vectors, Eulerian polynomials and stable polytopes of graphs. Electron. J. Combin. 11(2), #R6 (2004/06)Google Scholar
  2. 2.
    Bruns, W., Ichim, B., Römer, T., Sieg, R., Söger, C.: Normaliz: algorithms for rational cones and affine monoids. Available at https://www.normaliz.uni-osnabrueck.de
  3. 3.
    Bruns W., Römer T.: h-Vectors of Gorenstein polytopes. J. Combin. Theory Ser. A 114(1), 65–76 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Hibi T.: Gröbner Bases: Statistics and Software Systems. Springer, Tokyo (2013)CrossRefMATHGoogle Scholar
  5. 5.
    De Negri E., Hibi T.: Gorenstein algebras of Veronese type. J. Algebra 193(2), 629–639 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Nemhauser G.L., Trotter L.E. Jr.: Properties of vertex packing and independence system polyhedra. Math. Program. 6, 48–61 (1974)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ohsugi H., Hibi T.: Special simplices and Gorenstein toric rings. J. Combin. Theory Ser. A 113(4), 718–725 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Schepers J., Van Langenhoven L.: Unimodality questions for integrally closed lattice polytopes. Ann. Combin. 17(3), 571–589 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Schrijver A.: Theory of Linear and Integer Programming. John Wiley & Sons, Ltd., Chichester (1986)MATHGoogle Scholar
  10. 10.
    Stanley, R.P.: Enumerative Combinatorics. Vol. 1. Second Edition. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA (1986)Google Scholar
  11. 11.
    Stanley R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Stapledon A.: Inequalities and Ehrhart \({\delta}\)-vectors. Trans. Amer. Math. Soc. 361(10), 5615–5626 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Steingrímsson E.: A decomposition of 2-weak vertex-packing polytopes. Discrete Comput. Geom. 12(4), 465–479 (1994)MathSciNetCrossRefMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversitySuitaJapan
  2. 2.Department of Mathematical Sciences, School of Science and TechnologyKwansei Gakuin UniversitySandaJapan

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