# Finding a Hamilton cycle fast on average using rotations and extensions

## TL;DR

This paper introduces an efficient algorithm called CRE that finds Hamilton cycles in random graphs with optimal expected running time, improving upon previous methods for certain probability ranges.

## Contribution

The paper presents CRE, an algorithm that finds Hamilton cycles in random graphs with optimal expected time, advancing the state of the art in average-case complexity.

## Key findings

- Expected running time is (1+o(1))n/p for p ≥ 70n^{-1/2}
- Algorithm either finds a Hamilton cycle or confirms its absence
- Improves upon previous algorithms by Gurevich, Shelah, and Thomason

## Abstract

We present an algorithm CRE, which either finds a Hamilton cycle in a graph $G$ or determines that there is no such cycle in the graph. The algorithm's expected running time over input distribution $G\sim G(n,p)$ is $(1+o(1))n/p$, the optimal possible expected time, for $p=p(n) \geq 70n^{-\frac{1}{2}}$. This improves upon previous results on this problem due to Gurevich and Shelah, and to Thomason.