A note on long cycles in sparse random graphs

TL;DR

This paper investigates the asymptotic behavior of the longest cycle in sparse Erdős–Rényi random graphs and establishes a continuous limiting function for the cycle length ratio for sufficiently large average degree.

Contribution

It extends previous results to smaller average degrees and determines the probability of containing cycles of all lengths between shortest and longest for large graphs.

Findings

Existence of a continuous function f(c) for the ratio of longest cycle length for c ≥ 20.

Convergence of L_{c,n}/n to f(c) almost surely.

Asymptotic probability of containing cycles of all intermediate lengths.

Abstract

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