I just came across the notion of [ends][1] of a space, and I wonder if the following are equivalent for $G$ a locally finite connected graph:

 1. There exists an infinite path $v_1,v_2,\dots$ in $G$ which hits every vertex at least once but not infinitely many times
 2. $G$ is 1-ended


  [1]: https://en.wikipedia.org/wiki/End_(topology)