Small Complete Minors Above the Extremal Edge Density

TL;DR

This paper strengthens a classical graph theory result by replacing high connectivity with vertex expansion and proves a near-optimal bound on the size of minors in graphs with many edges.

Contribution

It extends Mader's theorem by using vertex expansion and establishes a tight bound on the size of complete minors in dense graphs.

Findings

Graphs with high average degree contain highly connected subgraphs.

Graphs with many edges contain small complete minors of logarithmic size.

The bound on minor size is tight up to a loglog n factor.

Abstract

A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader's result by replacing the notion of high connectivity by the notion of vertex expansion. Another well known result in graph theory states that for every integer t there is a smallest real c(t) so that every n-vertex graph with c(t)n edges contains a K_t-minor. Fiorini, Joret, Theis and Wood conjectured that if an n-vertex graph G has (c(t)+\epsilon)n edges then G contains a K_t-minor of order at most C(\epsilon)log n. We use our extension of Mader's theorem to prove that such a graph G must contain a K_t-minor of order at most C(\epsilon)log n loglog n. Known constructions of graphs with high girth show that this result is tight up to the loglog n factor.