# Hamilton cycles and perfect matchings in the KPKVB model

## TL;DR

This paper investigates the presence of Hamilton cycles and perfect matchings in a hyperbolic random graph model, revealing conditions under which these structures almost surely exist or do not, based on model parameters.

## Contribution

It provides new probabilistic thresholds for the existence of Hamilton cycles and perfect matchings in the KPKVB hyperbolic graph model.

## Key findings

- Hamilton cycles appear with high probability for large average degree.
- Perfect matchings are absent with high probability for small average degree.
- Results depend on the parameters controlling degree distribution and average degree.

## Abstract

In this paper we consider the existence of Hamilton cycles and perfect matchings in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, "short distances" and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes $n$, which we think of as going to infinity, and $\alpha, \nu > 0$, which we think of as constant. Roughly speaking $\alpha$ controls the power law exponent of the degree sequence and $\nu$ the average degree. Here we show that for every $\alpha < 1/2$ and $\nu=\nu(\alpha)$ sufficiently small, the model does not contain a perfect matching with high probability, whereas for every $\alpha < 1/2$ and $\nu=\nu(\alpha)$ sufficiently large, the model contains a Hamilton cycle with high probability.