Perfect matchings and Hamiltonian cycles in the preferential attachment model

TL;DR

This paper investigates the existence of perfect matchings and Hamiltonian cycles in preferential attachment graphs, establishing thresholds for their almost sure existence and comparing with a simpler uniform attachment model.

Contribution

It provides new thresholds for perfect matchings and Hamiltonian cycles in preferential attachment graphs and extends results from the uniform attachment model.

Findings

Perfect matchings exist asymptotically almost surely for m ≥ 1260.

Hamiltonian cycles exist asymptotically almost surely for m ≥ 29500.

Analysis addresses dependencies and non-independence of edges in the preferential attachment model.

Abstract

In this paper, we study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with $m$ random vertices selected with probabilities proportional to their current degrees. (Constant $m$ is the only parameter of the model.) We prove that if $m \ge 1{,}260$, then asymptotically almost surely there exists a perfect matching. Moreover, we show that there exists a Hamiltonian cycle asymptotically almost surely, provided that $m \ge 29{,}500$. One difficulty in the analysis comes from the fact that vertices establish connections only with vertices that are "older" (i.e. are created earlier in the process). However, the main obstacle arises from the fact that edges in the preferential attachment model are not generated independently. In view of that, we also consider a simpler setting---sometimes called uniform attachment---in which vertices are added one by one and each vertex connects to $m$ older vertices selected uniformly at random and independently of all other choices. We first investigate the existence of perfect matchings and Hamiltonian cycles in the uniform attachment model, and then extend the argument to the preferential attachment version.