Large highly connected subgraphs in graphs with linear average degree

TL;DR

This paper improves a classical bound on the existence of highly connected subgraphs in graphs with linear average degree, establishing a sharper threshold that is proven to be optimal.

Contribution

The authors refine Mader's theorem by lowering the average degree bound from 4k to 3+1/3 k, providing a tight bound for the existence of large, highly connected subgraphs.

Findings

The new bound of 3+1/3 k is sharp.

Graphs with average degree at least 3+1/3 k contain larger highly connected subgraphs.

The result tightens the relationship between average degree and connectivity in graphs.

Abstract

In 1972 Mader proved that every graph with average degree at least $4k$ has a $(k+1)$-connected subgraph with more than $2k$ vertices. We improve this bound by showing that the constant $4$ can be replaced by $3+\frac{1}{3}$; this bound is sharp.