TITLE: Non-isomorphic graphs with bijective graph homomorphisms in both directions between them
QUESTION [12 upvotes]: Are there simple, undirected graphs $G, H$ that are non-isomorphic, but there exist graph homomorphisms $f_1: G\to H$ and $f_2: H\to G$ which are bijective set-maps $V(G)\rightarrow V(H)$ and $V(H)\rightarrow V(G)$?
Notes. 

By the argument in Tobias Fritz's comment below, $G, H$ have to be infinite.
As suggested by a commenter, one should make it unambiguously clear that here, 'simple, undirected graph'='irreflexive symmetric binary relation on a set'.

REPLY [3 votes]: I got here via this question and I thought it may be worth sharing the following uncountable family of locally finite examples:
Pick an arbitrary set $S \subseteq \mathbb Z$. The vertex set of the graph $G_S$ is $\mathbb Z \times \mathbb Z$. For the edge set take all "vertical" edges from $(m,n)$ to $(m,n+1)$, horizontal edges $(m,n)$ to $(m+1,n)$ for $n < 0$, and horizontal edges $(s,0)$ to $(s+1,0)$ for $s \in S$; note that no horizontal edges are attached to $(m,n)$ for $n>0$.
As long as $S$ and $S'$ are not shifts/reflections of one another, the graphs $G_S$ and $G_{S'}$ are non-isomorphic. The map $f \colon G_S \to G_{S'}$ can be taken as $(m,n) \mapsto (m,n-1)$.