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arXiv · 2411.08339

Expected degrees in random plane graphs

Neely Lovvorn·Oscar Murillo-Espinoza·Adam Sheffer
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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.

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Neely Lovvorn, Oscar Murillo-Espinoza, Adam Sheffer. 2024-11-13. Expected degrees in random plane graphs. https://arxiv.org/abs/2411.08339

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