A note on long cycles in sparse random graphs

Let $L_{c,n}$ denote the size of the longest cycle in $G(n,c\mathrm{/}n)$, $c>1$ constant. We show that there exists a continuous function $f(c)$ such that $L_{c,n}\mathrm{/}n\to f(c)$ a.s. for $c\ge 20$, thus extending a result of the author and Frieze to smaller values of $c$. Thereafter, for $c\ge 20$, we determine the limit of the probability that $G(n,c\mathrm{/}n)$ contains cycles of every length between the length of its shortest and its longest cycles as $n\to \mathrm{\infty }$.