Expected degrees in random plane graphs
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929589541928960 |
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| author | Lovvorn, Neely Murillo-Espinoza, Oscar Sheffer, Adam |
| author_facet | Lovvorn, Neely Murillo-Espinoza, Oscar Sheffer, Adam |
| contents | 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 $n/\sqrt{πi}$
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08339 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Expected degrees in random plane graphs Lovvorn, Neely Murillo-Espinoza, Oscar Sheffer, Adam Combinatorics 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 $n/\sqrt{πi}$ 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. |
| title | Expected degrees in random plane graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.08339 |