Small Subgraphs with Large Average Degree

Abstract

In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number s>2$$s>2$$, we prove that every graph on n vertices with average degree d≥s$$d\ge s$$ contains a subgraph of average degree at least s on at most nd-ss-2(logd)Os(1)$$nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}$$ 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 n1-2s+ε$$n^{1-\frac{2}{s}+\varepsilon }$$ contains a subgraph of average degree at least s on Oε,s(1)$$O_{\varepsilon ,s}(1)$$ vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte.