The concept of neighborhood maps was looked at in a previous question.

Let $G= (V,E)$ be a simple, undirected graph. For $v\in V$ we set $N(v) = \{w\in V: \{v,w\} \in E\}$. Note that we always have $v\notin N(v)$. A function $f:V\to V$ is said to be a *neighborhood map* if $f(v)\in N(v)$ for all $v\in V$.

Does there exist a graph $G=(V,E)$ that admits an injective neighborhood map, but not a bijective one?

(Clearly, if such a graph exists, it is necessarily infinite.)