Small subgraphs with large average degree

TL;DR

This paper proves optimal bounds on the size of small dense subgraphs in graphs with high average degree, resolving two longstanding conjectures in graph theory.

Contribution

It establishes tight bounds on small dense subgraphs, confirming conjectures by Feige, Wagner, and Verstra"ete, and advances understanding of graph density properties.

Findings

Every graph with average degree at least d contains a small dense subgraph with degree s.

Optimal bounds are achieved up to polylogarithmic factors.

Results resolve two major conjectures in the field.

Abstract

In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $s>2$, we prove that every graph on $n$ vertices with average degree at least $d$ contains a subgraph of average degree at least $s$ on at most $nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}$ vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with $n$ vertices and average degree at least $n^{1-\frac{2}{s}+\varepsilon}$ contains a subgraph of average degree at least $s$ on $O_{\varepsilon,s}(1)$ vertices, which is also optimal up to the constant hidden in the $O(.)$ notation, and resolves a conjecture of Verstra\"ete.