On the cover time of the emerging giant

TL;DR

This paper determines the asymptotic cover time of the giant component in a sparse random graph as it emerges, refining previous bounds to an exact asymptotic expression.

Contribution

It establishes the precise asymptotic cover time for the giant component in the critical window of the Erdős–Rényi graph, improving upon prior approximate results.

Findings

Cover time asymptotic to n log^2 N

Refines previous bounds to exact asymptotics

Valid for sparse graphs with epsilon tending to zero

Abstract

Let $p=\frac{1+\varepsilon}{n}$. It is known that if $N=\varepsilon^3n\to\infty$ then w.h.p. $G_{n,p}$ has a unique giant largest component. We show that if in addition, $\varepsilon=\varepsilon(n)\to 0$ then w.h.p. the cover time of $G_{n,p}$ is asymptotic to $n\log^2N$; previously Barlow, Ding, Nachmias and Peres had shown this up to constant multiplicative factors.