On Small Separations in Cayley Graphs

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.