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 $G(n,(1+\epsilon)/n)$ that holds w.h.p. for all $\epsilon=\epsilon(n)$ such that $\epsilon^3n\to \infty$. In the case $\epsilon\leq \epsilon_0$ for some sufficiently small constant $\epsilon_0$, this bound is equal to $1.581\epsilon^2n$ which improves upon the current best lower bound of $4\epsilon^2n/3$ due to Luczak.