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ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH

Year 2021, , 269 - 284, 17.07.2021
https://doi.org/10.24330/ieja.969935

Abstract

In this paper,
we give a criterion of the Gorenstein property of
the Ehrhart ring of the stable set polytope of
an h-perfect graph:
the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if
(1)
sizes of maximal cliques are constant (say $n$) and
(2)
(a)
$n=1$,
(b)
$n=2$ and there is no odd cycle without chord and length at least 7 or
(c)
$n\geq 3$ and there is no odd cycle without chord and length at least 5.

References

  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
  • V. Chvatal, On certain polytopes associated with graphs, J. Combinatorial Theory Ser. B 18(2), (1975), 138-154.
  • R. M. Fossum, The Divisor Class Group of a Krull Domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York- Heidelberg, 1973.
  • J. Herzog, T. Hibi and D. I. Stamate, The trace of the canonical module, Israel J. Math., 233(1) (2019), 133-165.
  • T. Hibi and A. Tsuchiya, Odd cycles and Hilbert functions of their toric rings, Mathematics, 8(1) (2020), 22.
  • M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials and polytopes, Ann. of Math., 96 (1972), 318-337.
  • M. Miyazaki, On the canonical ideal of the Ehrhart ring of the chain polytope of a poset, J. Algebra, 541 (2020), 1-34.
  • H. Ohsugi and T. Hibi, Special simplices and Gorenstein toric rings, J. Combin. Theory Ser. A, 113(4) (2006), 718-725.
  • N. Sbihi and J.-P. Uhry, A class of h-perfect graphs, Discrete Math., 51(2) (1984), 191-205.
  • R. P. Stanley, Hilbert functions of graded algebras, Advances in Math., 28(1) (1978), 57-83.
  • R. P. Stanley, Two poset polytopes, Discrete Comput. Geom., 1(1) (1986), 9-23.
Year 2021, , 269 - 284, 17.07.2021
https://doi.org/10.24330/ieja.969935

Abstract

References

  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
  • V. Chvatal, On certain polytopes associated with graphs, J. Combinatorial Theory Ser. B 18(2), (1975), 138-154.
  • R. M. Fossum, The Divisor Class Group of a Krull Domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York- Heidelberg, 1973.
  • J. Herzog, T. Hibi and D. I. Stamate, The trace of the canonical module, Israel J. Math., 233(1) (2019), 133-165.
  • T. Hibi and A. Tsuchiya, Odd cycles and Hilbert functions of their toric rings, Mathematics, 8(1) (2020), 22.
  • M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials and polytopes, Ann. of Math., 96 (1972), 318-337.
  • M. Miyazaki, On the canonical ideal of the Ehrhart ring of the chain polytope of a poset, J. Algebra, 541 (2020), 1-34.
  • H. Ohsugi and T. Hibi, Special simplices and Gorenstein toric rings, J. Combin. Theory Ser. A, 113(4) (2006), 718-725.
  • N. Sbihi and J.-P. Uhry, A class of h-perfect graphs, Discrete Math., 51(2) (1984), 191-205.
  • R. P. Stanley, Hilbert functions of graded algebras, Advances in Math., 28(1) (1978), 57-83.
  • R. P. Stanley, Two poset polytopes, Discrete Comput. Geom., 1(1) (1986), 9-23.
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mitsuhiro Mıyazakı This is me

Publication Date July 17, 2021
Published in Issue Year 2021

Cite

APA Mıyazakı, M. (2021). ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH. International Electronic Journal of Algebra, 30(30), 269-284. https://doi.org/10.24330/ieja.969935
AMA Mıyazakı M. ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH. IEJA. July 2021;30(30):269-284. doi:10.24330/ieja.969935
Chicago Mıyazakı, Mitsuhiro. “ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH”. International Electronic Journal of Algebra 30, no. 30 (July 2021): 269-84. https://doi.org/10.24330/ieja.969935.
EndNote Mıyazakı M (July 1, 2021) ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH. International Electronic Journal of Algebra 30 30 269–284.
IEEE M. Mıyazakı, “ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH”, IEJA, vol. 30, no. 30, pp. 269–284, 2021, doi: 10.24330/ieja.969935.
ISNAD Mıyazakı, Mitsuhiro. “ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH”. International Electronic Journal of Algebra 30/30 (July 2021), 269-284. https://doi.org/10.24330/ieja.969935.
JAMA Mıyazakı M. ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH. IEJA. 2021;30:269–284.
MLA Mıyazakı, Mitsuhiro. “ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH”. International Electronic Journal of Algebra, vol. 30, no. 30, 2021, pp. 269-84, doi:10.24330/ieja.969935.
Vancouver Mıyazakı M. ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH. IEJA. 2021;30(30):269-84.