Abstract
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.
Highlights
Let P ⊂ Rn be an integral convex polytope, i.e., a convex polytope whose vertices have integer coordinates, and let P ∩ Zn = {a1, . . . , am }
IP is generated by homogeneous binomials and any reduced Gröbner basis of IP consists of homogeneous binomials; see [1]
The purpose of this paper is to study such properties of toric rings and ideals of stable set polytopes of simple graphs
Summary
The purpose of this paper is to study such properties of toric rings and ideals of stable set polytopes of simple graphs. The toric ideals IQG of the stable set polytope QG of a simple graph G. G of stability number two, QG is normal if and only if the complement of G satisfies the “odd cycle condition” (Theorem 1) Using this criterion, we construct an infinite family of normal stable set polytopes without regular unimodular triangulations (Theorem 2). Using the results on normality, generators, and Gröbner bases, we present an infinite family of non-normal stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases (Theorem 4)
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