Small separations in vertex transitive graphs

Abstract

Let $k$ be an integer. We prove a rough structure theorem for separations of order at most $k$ in finite and infinite vertex transitive graphs. Let $G=(V,E)$ be a vertex transitive graph, let $A\subseteq V$ be a finite vertex-set with $|A|\le |V|/2$ and $|\{v \in V \setminus A : {$u \sim v$forsome$u \in A$} \}|\le k$. We show that whenever the diameter of $G$ is at least $31(k+1{)}^2$, either $|A|\le 2k^3+k^2$, or $G$ has a ring-like structure (with bounded parameters), and $A$ is efficiently contained in an interval. This theorem may be viewed as a rough characterization, generalizing an earlier result of Tindell, and has applications to the study of product sets and expansion in groups.

28 pages