The Farey graph is uniquely determined by its connectivity

TL;DR

This paper proves that the Farey graph is uniquely characterized among infinitely edge-connected graphs by its property that any two vertices can be separated by finitely many edges, up to minor-equivalence.

Contribution

It establishes the uniqueness of the Farey graph based on its connectivity properties, providing a new characterization in graph theory.

Findings

Farey graph is the only minor-minimal infinitely edge-connected graph with finitely separable vertex pairs.

The result characterizes the Farey graph through its connectivity and separation properties.

The proof relies on properties of minor-minimality and edge connectivity in infinite graphs.

Abstract

We show that, up to minor-equivalence, the Farey graph is the unique minor-minimal graph that is infinitely edge-connected but such that every two vertices can be finitely separated.