Perfect matchings and Hamilton cycles in uniform attachment graphs

TL;DR

This paper investigates the presence of perfect matchings and Hamilton cycles in uniform attachment graphs, showing that such structures appear with high probability when the parameter k is sufficiently large.

Contribution

It improves previous results by establishing new thresholds for the existence of perfect matchings and Hamilton cycles in uniform attachment graphs.

Findings

Perfect matchings exist with high probability for k ≥ 5.

Hamilton cycles exist with high probability for k ≥ 13.

New expansion properties are identified to support these results.

Abstract

We study Hamilton cycles and perfect matchings in a uniform attachment graph. In this random graph, vertices are added sequentially, and when a vertex $t$ is created, it makes $k$ independent and uniform choices from $\{1,\dots,t-1\}$ and attaches itself to these vertices. Improving the results of Frieze, P\'erez-Gim\'enez, Pra\l{}at and Reiniger (2019), we show that, with probability approaching 1 as $n$ tends to infinity, a uniform attachment graph on $n$ vertices has a perfect matching for $k \ge 5$ and a Hamilton cycle for $k\ge 13$. One of the ingredients in our proofs is the identification of a subset of vertices that is least likely to expand, which provides us with better expansion rates than the existing ones.