A simple, undirected graph $G=(V,E)$ is said to be *vertex-transitive* if for all $a,b\in V$ there is a graph isomorphism $\varphi:G\to G$ such that $\varphi(a) = b$.


If $G = (\omega, E)$ is vertex-transitive and connected, is there a bijection $p:\mathbb{Z}\to \omega$ such that $\{p(k), p(k+1)\} \in E$ for all $k\in\mathbb{Z}$?



Hamiltonian $\mathbb{Z}$-paths in connected countably infinite vertex-transitive graphs
If $G$ is a regular tree of degree $d \geq 3$, there's clearly no such $p$.


