An improved lower bound on the length of the longest cycle in random graphs

Abstract

We provide a new lower bound on the length of the longest cycle of the binomial random graph $G(n,(1+ε)/n)$ that holds w.h.p. for all $ε=ε(n)$ such that $ε^3n\to \infty$. In the case $ε\leq ε_0$ for some sufficiently small constant $ε_0$, this bound is equal to $1.581ε^2n$ which improves upon the current best lower bound of $4ε^2n/3$ due to Luczak.