Matching polytopes, Gorensteinness, and the integer decomposition property

TL;DR

This paper characterizes graphs with Gorenstein matching polytopes and proves that all such polytopes have the integer decomposition property, with a case study on wheel graphs.

Contribution

It provides a complete characterization of graphs with Gorenstein matching polytopes and establishes the integer decomposition property for all Gorenstein matching polytopes.

Findings

Gorenstein matching polytopes are characterized by specific graph properties.

All Gorenstein matching polytopes have the integer decomposition property.

Wheel graph matching polytopes are not Gorenstein but have the integer decomposition property.

Abstract

The matching polytope of a graph $G$ is the convex hull of the indicator vectors of the matchings on $G$. We characterize the graphs whose associated matching polytopes are Gorenstein, and then prove that all Gorenstein matching polytopes possess the integer decomposition property. As a special case study, we examine the matching polytopes of wheel graphs and show that they are not Gorenstein, but do possess the integer decomposition property.