The evolution of random graphs on surfaces

TL;DR

This paper studies the properties of random graphs on surfaces with bounded genus, revealing how their structure changes with the ratio of edges to vertices, including component types, degrees, and face sizes.

Contribution

It provides a detailed asymptotic analysis of random graphs on surfaces, highlighting phase transitions based on the edge-to-vertex ratio and extending understanding beyond planar graphs.

Findings

Non-planar components become unlikely as n grows

Planar subgraphs appear with high probability when m/n > 1

Maximum degree scales as Θ(ln n) when m/n > 1

Abstract

For integers $g,m \geq 0$ and $n>0$, let $S_{g}(n,m)$ denote the graph taken uniformly at random from the set of all graphs on $\{1,2, \ldots, n\}$ with exactly $m=m(n)$ edges and with genus at most $g$. We use counting arguments to investigate the components, subgraphs, maximum degree, and largest face size of $S_{g}(n,m)$, finding that there is often different asymptotic behaviour depending on the ratio $\frac{m}{n}$. In our main results, we show that the probability that $S_{g}(n,m)$ contains any given non-planar component converges to $0$ as $n \to \infty$ for all $m(n)$; the probability that $S_{g}(n,m)$ contains a copy of any given planar graph converges to $1$ as $n \to \infty$ if $\liminf \frac{m}{n} > 1$; the maximum degree of $S_{g}(n,m)$ is $\Theta (\ln n)$ with high probability if $\liminf \frac{m}{n} > 1$; and the largest face size of $S_{g}(n,m)$ has a threshold around $\frac{m}{n}=1$ where it changes from $\Theta (n)$ to $\Theta (\ln n)$ with high probability.