Coarse selectors of graphs

We consider a connected graph $\mathrm{\Gamma }$ as a coarse space and prove that $\mathrm{\Gamma }$ admits a 2-selector if and only if $\mathrm{\Gamma }$ 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.