Non-Gorenstein locus and almost Gorenstein property of the Ehrhart ring of the stable set polytope of a cycle graph

TL;DR

This paper investigates the algebraic properties of the Ehrhart ring of the stable set polytope of a cycle graph, specifically its non-Gorenstein locus and almost Gorenstein status, confirming a conjecture by Hibi and Tsuchiya.

Contribution

It characterizes the non-Gorenstein locus of the Ehrhart ring and proves that it is almost Gorenstein, validating a conjecture by Hibi and Tsuchiya.

Findings

The Ehrhart ring of the stable set polytope of a cycle graph is almost Gorenstein.

The non-Gorenstein locus of the Ehrhart ring is explicitly described.

The conjecture of Hibi and Tsuchiya is confirmed.

Abstract

Let $R$ be the Ehrhart ring of the stable set polytope of a cycle graph which is not Gorenstein. We describe the non-Gorenstein locus of $\mathrm{Spec} R$. Further, we show that $R$ is almost Gorenstein. Moreover, we show that the conjecture of Hibi and Tsuchiya is true.