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.