Small separations in vertex transitive graphs

TL;DR

This paper establishes a structural theorem for small separations in vertex transitive graphs, revealing conditions under which such graphs exhibit ring-like structures or small vertex sets, with implications for group theory and graph expansion.

Contribution

It generalizes previous results by characterizing the structure of small separations in finite and infinite vertex transitive graphs, especially those with large diameter.

Findings

Graphs with large diameter and small separations are either small or have a ring-like structure.

The theorem applies to both finite and infinite vertex transitive graphs.

Implications for product sets and expansion in groups.

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$ for some $u \in A$} \}|\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.