Coarse selectors of graphs

TL;DR

This paper characterizes when a connected graph admits a 2-selector based on its coarse equivalence class, linking coarse geometry with linear orderings in metric spaces.

Contribution

It provides a complete characterization of graphs with 2-selectors in coarse geometry, connecting boundedness, coarse equivalence to , , and applications to geodesic metric spaces.

Findings

Graphs admit 2-selectors iff they are bounded or coarsely equivalent to  or .

Application to geodesic metric spaces with compatible linear orders.

Provides a classification linking coarse structures and linear orderings.

Abstract

We consider a connected graph $\Gamma$ as a coarse space and prove that $\Gamma$ admits a 2-selector if and only if $\Gamma$ is either bounded or coarsely equivalent to $\mathbb{N}$ or $\mathbb{Z}$. We apply this result to geodesic metric spaces admitting linear orders compatible with coarse structures.