Random graphs embeddable in order-dependent surfaces

TL;DR

This paper studies the properties of random graphs embeddable in surfaces of varying genus, revealing phase transitions in connectivity and component structure depending on the genus function.

Contribution

It characterizes the typical structure and phase transitions of random graphs embeddable in order-dependent surfaces, extending known results for planar and Erdős–Rényi graphs.

Findings

At most one non-planar component with high probability.

Connectivity probability transitions depending on genus growth rate.

Results hold for both orientable and non-orientable surfaces.

Abstract

Given a `genus' function $g=g(n)$, we let $\mathcal{E}^g$ be the class of all graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it is embeddable in a surface of Euler genus at most $g(n)$. Let the random graph $R_n$ be sampled uniformly from the graphs in $\mathcal{E}^g$ on vertex set $[n]=\{1,\ldots,n\}$. Observe that if $g(n)$ is 0 then $R_n$ is a random planar graph, and if $g(n)$ is sufficiently large then $R_n$ is a binomial random graph $G(n,\tfrac12)$. We investigate typical properties of $R_n$. We find that for \emph{every} genus function $g$, with high probability at most one component of $R_n$ is non-planar. In contrast, we find a transition for example for connectivity: if $g$ is non-decreasing and $g(n) = O(n/\log n)$ then $\liminf_{n \to \infty} \mathbb{P}(R_n \mbox{ is connected}) < 1$, and if $g(n) \gg n$ then with high probability $R_n$ is connected. These results also hold when we consider orientable and non-orientable surfaces separately. We also investigate random graphs sampled uniformly from the `hereditary part' or the `minor-closed' part of $\mathcal{E}^g$, and briefly consider corresponding results for unlabelled graphs.