An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three

TL;DR

This paper presents an almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three, especially effective when the number of edges is proportional to the number of vertices.

Contribution

The paper introduces a new efficient algorithm that finds Hamilton cycles in sparse random graphs with high probability, improving computational complexity for this class of graphs.

Findings

Algorithm runs in O(n^{1+o(1)}) time for large enough c

Successfully finds Hamilton cycles with high probability in the specified model

Applicable to graphs with minimum degree at least three and linear edge count

Abstract

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph $G=\gc$. In this model $G$ is drawn uniformly from graphs with vertex set $[n]$, $m$ edges and minimum degree at least three. We focus on the case where $m=cn$ for constant $c$. If $c$ is sufficiently large then our algorithm runs in $O(n^{1+o(1)})$ time and succeeds w.h.p.