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

We provide a new lower bound on the length of the longest cycle of the binomial random graph $G(n,(1+\epsilon )\mathrm{/}n)$ that holds w.h.p. for all $\epsilon =\epsilon (n)$ such that ${\epsilon }^{3}n\to \mathrm{\infty }$. In the case $\epsilon \le {\epsilon }_0$ for some sufficiently small constant ${\epsilon }_0$, this bound is equal to $1.581{\epsilon }^{2}n$ which improves upon the current best lower bound of $4{\epsilon }^{2}n\mathrm{/}3$ due to Luczak.