I just came across the notion of ends of a space, and I wonder if the following are equivalent for $G$ a locally finite 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

I just came across the notion of ends 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