A greedy algorithm for finding a large 2-matching on a random cubic graph

TL;DR

This paper analyzes a greedy algorithm for finding large 2-matchings in random cubic graphs, showing it typically produces a 2-matching with a small number of components proportional to n^{1/5}.

Contribution

It provides a probabilistic analysis of a greedy algorithm's effectiveness in large random cubic graphs, establishing bounds on the number of components in the resulting 2-matching.

Findings

The algorithm outputs a 2-matching with about n^{1/5} components.

The analysis applies with high probability to random 3-regular graphs.

The size of the 2-matching approaches the maximum possible in such graphs.

Abstract

A 2-matching of a graph $G$ is a spanning subgraph with maximum degree two. The size of a 2-matching $U$ is the number of edges in $U$ and this is at least $n-\k(U)$ where $n$ is the number of vertices of $G$ and $\k$ denotes the number of components. In this paper, we analyze the performance of a greedy algorithm \textsc{2greedy} for finding a large 2-matching on a random 3-regular graph. We prove that with high probability, the algorithm outputs a 2-matching $U$ with $\k(U) = \tilde{\Theta}\of{n^{1/5}}$.