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 $$d\ge s$$ d ≥ s contains a subgraph of average degree at least s on at most $$nd^{-\frac{s}{s-2}}(\log d)^{O_s(1)}$$ n d - 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 $$n^{1-\frac{2}{s}+\varepsilon }$$ n 1 - 2 s + ε contains a subgraph of average degree at least s on $$O_{\varepsilon ,s}(1)$$ O ε , s ( 1 ) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte.