Expected degrees in random plane graphs

TL;DR

This paper analyzes the expected degrees and isolated vertices in random plane graphs on point sets, providing bounds and constructions that reveal typical graph properties.

Contribution

It introduces a novel analysis method using cross-graph charging schemes to estimate expected degrees and isolated vertices in random plane graphs.

Findings

Expected isolated vertices less than n/10.18 in random plane graphs.

Constructed point set with about n/23.32 expected isolated vertices.

Expected number of vertices with degree i less than n/√(πi).

Abstract

We prove that, for every set of $n$ points $\mathcal{P}$ in $\mathbb{R}^2$, a random plane graph drawn on $\mathcal{P}$ is expected to contain less than $n/10.18$ isolated vertices. In the other direction, we construct a point set where the expected number of isolated vertices in a random plane graph is about $n/23.32$. For $i\ge 1$, we prove that the expected number of vertices of degree $i$ is always less than $n/\sqrt{\pi i}$ Our analysis is based on cross-graph charging schemes. That is, we move charge between vertices from different plane graphs of the same point set. This leads to information about the expected behavior of a random plane graph.