Gorenstein on the punctured spectrum and nearly Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph

Abstract

In this paper, we give a criterion of the nearly 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 with connected components G (1) , . . . , G (ℓ) is nearly Gorenstein if and only if (1) for each i, the Ehrhart ring of the stable set polytope of G (i) is Gorenstein and (2) |ω(G (i) ) − ω(G (j) )| ≤ 1 for any i and j, where ω(G (i) ) is the clique number of G (i) .We also show that the Segre product of Cohen-Macaulay graded rings with linear non-zerodivisor which are Gorenstein on the punctured spectrum is also Gorenstein on the punctured spectrum if all but one rings are standard graded. MSC: 13H10,

Nearly Gorenstein Polytopes

“…This family of polytopes includes many subfamilies of polytopes which arise in combinatorics, such as order polytopes of posets and base polytopes from graphic matroids. Previous work has studied nearly Gorensteinness of Hibi rings [8] and of Ehrhart rings of stable set polytopes arising from perfect graphs [14,18]. The main result of this section generalises these previous results by characterising a large class of nearly Gorenstein (0, 1)-polytopes: Theorem 1.4 (Theorem 4.2).…”

Nearly Gorenstein Polytopes

“…This family of polytopes includes many subfamilies of polytopes which arise in combinatorics, such as order polytopes of posets and base polytopes from graphic matroids. Previous work has studied nearly Gorensteinness of Hibi rings [8] and of Ehrhart rings of stable set polytopes arising from perfect graphs [14,18]. The main result of this section generalises these previous results by characterising a large class of nearly Gorenstein (0, 1)-polytopes: Theorem 4 (Theorem 34).…”

“…[21]). Moreover, the characterisation of nearly Gorenstein stable set polytopes of perfect graphs has been given in [14,18]. Let G be a perfect graph with connected components G 1 , .…”