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

## TL;DR

This paper investigates the asymptotic behavior of the longest cycle in sparse random graphs, establishing a limiting function for its normalized length as the graph size grows, especially for large average degrees.

## Contribution

It introduces a new limiting function for the longest cycle length in sparse random graphs and provides explicit formulas for initial polynomial coefficients.

## Key findings

- Longest cycle length converges to a function f(c) of the average degree c.
- For large c, the normalized longest cycle length approaches f(c).
- The same asymptotic applies to the longest path in the graph.

## Abstract

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 $f(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$ where $p_k$ is a polynomial in $k$. We are only able to explicitly give the values $p_1,p_2$, although we could in principle compute any $p_k$. We see immediately that the length of the longest path is also asymptotic to $f(c)n$ w.h.p.