It is known that for metric graphs the concepts of *Gromov's hyperbolicity* and *strictly positive* Cheeger constant are related. Let us first recall the definition of the Cheeger constant. Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For a collection of vertices $A \subseteq V(G)$, let $\partial A$ denote the collection of all edges going from a vertex in $A$ to a vertex outside of $A$,
$$
\partial A := \{ (x, y) \in E : x \in A, y \in V(G) \setminus A \}.
$$

Then the Cheeger constant of G, denoted $h(G)$, is defined as $$ h(G) := \inf \left\{ \frac{|\partial A|}{|A|} : A \subseteq V(G)\quad\text{and}\quad |A|<\infty \right\}. $$

However, neither Gromov's hyperbolicity nor strictly positive Cheeger constant implies the other. For instance, the graph $\mathbb{Z}$ is hyperbolic in the sense of Gromov but its Cheeger constant is zero. On the other hand, the Cayley graph of the group $\mathbb{F}_{2}\times\mathbb{F}_2$ has positive Cheeger constant but it is not hyperbolic since $\mathbb{Z}\times \mathbb{Z}$ sits as a subgroup.

Does there exist a *planar graph* with positive Cheeger constant but not hyperbolic?