Gorenstein cut polytopes

TL;DR

This paper characterizes Gorenstein cut polytopes of graphs, showing they are compressed and providing a graph-theoretic criterion involving minors and specific graph classes for when a cut polytope is Gorenstein.

Contribution

It explicitly characterizes Gorenstein cut polytopes of graphs and links their properties to graph minors and specific graph classes, extending the understanding of Gorenstein polytopes.

Findings

Gorenstein cut polytopes are compressed.

A cut polytope is Gorenstein iff the graph has no $K_5$-minor and is bipartite without large cycles or bridgeless chordal.

Provides a complete graph-theoretic characterization of Gorenstein cut polytopes.

Abstract

An integral convex polytope ${\mathcal P}$ is said to be Gorenstein if its toric ring $K[{\mathcal P}]$ is normal and Gorenstein. In this paper, Gorenstein cut polytopes of graphs are characterized explicitly. First, we prove that Gorenstein cut polytopes are compressed (i.e., all of whose reverse lexicographic triangulations are unimodular). Second, by applying Athanasiadis's theory for Gorenstein compressed polytopes, we show that a cut polytope of a graph $G$ is Gorenstein if and only if $G$ has no $K_5$-minor and $G$ is either a bipartite graph without induced cycles of length $\geq 6$ or a bridgeless chordal graph.