  We present two results on expansion of Cayley graphs. The first result
settles a conjecture made by DeVos and Mohar. Specifically, we prove that for
any positive constant $c$ there exists a finite connected subset $A$ of the
Cayley graph of $\mathbb{Z}^2$ such that $\frac{|\partial A|}{|A|}<
\frac{c}{depth(A)}$. This yields that there can be no universal bound for
$\frac{|\partial A|depth(A)}{|A|}$ for subsets of either infinite or finite
vertex transitive graphs. Let $X=(V,E)$ be the Cayley graph of a finitely
generated infinite group and $A\subset V$ finite such that $A\cup\partial A$ is
connected. Our second result is that if $|A|> 16|\partial A|^2$ then $X$ has a
ring-like structure.
