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

TL;DR

This paper establishes a new lower bound on the longest cycle length in certain random graphs, improving previous bounds and applicable under specific growth conditions of epsilon.

Contribution

It introduces a tighter lower bound for the longest cycle in binomial random graphs for a range of epsilon values, advancing theoretical understanding.

Findings

New lower bound of 1.581 epsilon^2 n for cycle length

Improves upon previous bound of 4 epsilon^2 n/3

Applicable when epsilon^3 n tends to infinity

Abstract

We provide a new lower bound on the length of the longest cycle of the binomial random graph $G(n,(1+\epsilon)/n)$ that holds w.h.p. for all $\epsilon=\epsilon(n)$ such that $\epsilon^3n\to \infty$. In the case $\epsilon\leq \epsilon_0$ for some sufficiently small constant $\epsilon_0$, this bound is equal to $1.581\epsilon^2n$ which improves upon the current best lower bound of $4\epsilon^2n/3$ due to Luczak.