# The facets of the spanning trees polytope

## TL;DR

This paper characterizes all facets of the spanning trees polytope of a graph by relating it to the facets of the bases polytope of an associated matroid, providing a comprehensive geometric description.

## Contribution

It provides a complete description of the facets of the spanning trees polytope using matroid theory, linking two important polyhedral structures.

## Key findings

- All facets of the spanning trees polytope are described.
- The description leverages the facets of the bases polytope of a matroid.
- The work unifies polyhedral descriptions through matroid properties.

## Abstract

Let $G=(V, E)$ be an undirected graph. The spanning trees polytope $P(G)$ is the convex hull of the characteristic vectors of all spanning trees of $G$. In this paper, we describe all facets of $P(G)$ as a consequence of the facets of the bases polytope $P(M)$ of a matroid $M$, i.e., the convex hull of the characteristic vectors of all bases of $M$.