# Hamilton cycles in random graphs with minimum degree at least 3: an improved analysis

## TL;DR

This paper improves the analysis of Hamilton cycle existence in random graphs with minimum degree at least 3, reducing the edge density threshold needed for Hamiltonicity from 10 to approximately 2.662, aligning with expansion bounds.

## Contribution

The paper provides an improved bound on the minimum edge density for Hamiltonicity in random graphs with minimum degree at least 3, lowering it from 10 to about 2.662.

## Key findings

- Hamilton cycles exist w.h.p. for c > 2.662 in G_{n,m}^{δ≥3}
- Reduced the lower bound for Hamiltonicity in such graphs
- Aligned the bound with expansion properties of Pósa sets

## Abstract

In this paper we consider the existence of Hamilton cycles in the random graph $G=G_{n,m}^{\delta\geq 3}$. This a random graph chosen uniformly from the set of graphs with vertex set $[n]$, $m$ edges and minimum degree at least 3. Our ultimate goal is to prove that if $m=cn$ and $c>3/2$ is constant then $G$ is Hamiltonian w.h.p. In an earlier paper the second author showed that $c\geq 10$ is sufficient for this and in this paper we reduce the lower bound to $c>2.662...$. This new lower bound is the same lower bound found in Frieze and Pittel \cite{FP} for the expansion of so-called P\'osa sets.