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Open AccessFeature PaperArticle

Toric Rings and Ideals of Stable Set Polytopes

1
Faculty of Engineering, Kitami Institute of Technology, Kitami, Hokkaido 090-8507, Japan
2
Department of Mathematical Sciences, School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo 669-1337, Japan
3
Department of Mathematics, College of Science, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(7), 613; https://doi.org/10.3390/math7070613
Received: 6 June 2019 / Revised: 6 July 2019 / Accepted: 8 July 2019 / Published: 10 July 2019
(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)
In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases. View Full-Text
Keywords: toric ideals; Gröbner bases; graphs; stable set polytopes toric ideals; Gröbner bases; graphs; stable set polytopes
MDPI and ACS Style

Matsuda, K.; Ohsugi, H.; Shibata, K. Toric Rings and Ideals of Stable Set Polytopes. Mathematics 2019, 7, 613.

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