I'm really curious about this question:
Let $G(V_G,E_G)$ and $H(V_H,E_H)$ be infinite (infinite!, not finite) graphs, such that $$|V_G|=|V_H|,$$ and let $f$, $g$ be functions $f: G\rightarrow H$, $g: H\rightarrow G$ such that $$\text{$f$ and $g$ are monomorphisms}.$$
By the given information, is it true to say that $G$ and $H$ are isomorphic?
From the given information, $G$ is isomorphic to a subgraph of $H$, and $H$ is isomorphic to a subgraph of $G$. If $f$ or $g$ is not an isomorphism, then $G$ is a proper subgraph of itself and $H$ is a proper subgraph of itself. This doesn't seem to create a contradiction.
No. For instance, let $G$ be a complete graph on an infinite set and let $H$ be a disjoint union of $G$ and one more vertex. Then there is an obvious monomorphism $G\to H$, and there is also a monomorphism $H\to G$ (just take any injection on vertices, and it is automatically a homomorphism since $G$ is complete). However, $G$ and $H$ are not isomorphic.