Largest sparse subgraphs of random graphs

TL;DR

This paper determines the asymptotic size of the largest sparse subgraphs in Erdős-Rényi random graphs, generalizing independence number results and using large deviations inequalities for precise estimates.

Contribution

It provides a precise asymptotic formula for the largest vertex subset inducing a subgraph with bounded average degree in G(n,p), extending previous results on independence numbers.

Findings

Asymptotic concentration on at most two points for fixed t and p

Generalization of independence number results in random graphs

Use of large deviations inequalities for bounds

Abstract

For the Erd\H{o}s-R\'enyi random graph G(n,p), we give a precise asymptotic formula for the size of a largest vertex subset in G(n,p) that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t = t(n) is not too large. In the case of fixed t and p, we find that this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.