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We discuss the length of the longest cycle in a sparse random graph$G_{n,p},p=c/n$. $c$ constant. We show that for large $c$ there is a function$f(c)$ such that $L_n(c)/n\to f(c)$ a.s. The function $f(c)=1-\sum_{k=1}^\inftyp_k(c)e^{-kc}$ where $p_k$ is a polynomial in $k$. We are only able toexplicitly give the values $p_1,p_2$, although we could in principle computeany $p_k$. We see immediately that the length of the longest path is alsoasymptotic to $f(c)n$ w.h.p.

A scaling limit for the length of the longest cycle in a sparse random graph