Finding and using expanders in locally sparse graphs

TL;DR

This paper proves that locally sparse graphs contain large expanding subgraphs, with implications for random graphs, graph minors, and positional games, and provides an algorithmic approach.

Contribution

It introduces the concept of $(c_1,c_2,eta)$-graphs and shows they contain large induced expanders, with an algorithmic proof and multiple applications.

Findings

Locally sparse graphs contain linearly sized expanders.

The proof is constructive and algorithmic.

Applications include random graphs and graph minor embedding.

Abstract

We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants $c_1>c_2>1$, $0<\alpha<1$, a graph $G$ on $n$ vertices is called a $(c_1,c_2,\alpha)$-graph if it has at least $c_1n$ edges, but every vertex subset $W\subset V(G)$ of size $|W|\le \alpha n$ spans less than $c_2|W|$ edges. We prove that every $(c_1,c_2,\alpha)$-graph with bounded degrees contains an induced expander on linearly many vertices. The proof can be made algorithmic. We then discuss several applications of our main result to random graphs, to problems about embedding graph minors, and to positional games.