On a Greedy 2-Matching Algorithm and Hamilton Cycles in Random Graphs with Minimum Degree at Least Three

TL;DR

This paper presents a simple greedy algorithm that efficiently finds a near-optimal 2-matching in certain random graphs, which is then used to construct Hamilton cycles with high probability.

Contribution

The paper introduces a straightforward greedy algorithm for 2-matching in random graphs with minimum degree three, enabling Hamilton cycle construction.

Findings

The greedy algorithm finds a 2-matching with O(log n) components.

The 2-matching can be transformed into a Hamilton cycle in sub-quadratic time.

The method works with high probability for graphs with c ≥ 15.

Abstract

We describe and analyse a simple greedy algorithm \2G\ that finds a good 2-matching $M$ in the random graph $G=G_{n,cn}^{\d\geq 3}$ when $c\geq 15$. A 2-matching is a spanning subgraph of maximum degree two and $G$ is drawn uniformly from graphs with vertex set $[n]$, $cn$ edges and minimum degree at least three. By good we mean that $M$ has $O(\log n)$ components. We then use this 2-matching to build a Hamilton cycle in $O(n^{1.5+o(1)})$ time \whp.