Three families of toric rings arising from posets or graphs with small class groups

TL;DR

This paper analyzes three types of toric rings from posets and graphs with small class groups, describing their structure, class groups, and conditions for low rank, revealing differences among isomorphic classes.

Contribution

It provides a detailed analysis of the class groups of Hibi rings, stable set rings, and edge rings with small rank, including characterizations for ranks 1 and 2.

Findings

Class groups of the three toric rings are torsionfree.

Characterizations of posets and graphs with class group rank 1 or 2.

Discussion of differences among isomorphic classes with small class groups.

Abstract

The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreover, we characterize the posets or graphs whose associated toric rings have rank $1$ or $2$. By using those characterizations, we discuss the differences of isomorphic classes of those toric rings with small class groups.