Maximal knotless graphs

A graph is maximal knotless if it is edge maximal for the property of knotless embedding in \(R^3\). We show that such a graph has at least \(\frac74 |V|\) edges, and construct an infinite family of maximal knotless graphs with \(|E| < \frac52|V|\). With the exception of \(|E| = 22\), we show tha...

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Bibliographic Details
Published in:arXiv.org 2021-02
Main Authors: Eakins, Lindsay, Fleming, Thomas, Mattman, Thomas W
Format: Article
Language:English
Subjects:
Apexes
Graph theory
Graphs
Online Access:Get full text
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Summary:A graph is maximal knotless if it is edge maximal for the property of knotless embedding in \(R^3\). We show that such a graph has at least \(\frac74 |V|\) edges, and construct an infinite family of maximal knotless graphs with \(|E| < \frac52|V|\). With the exception of \(|E| = 22\), we show that for any \(|E| \geq 20\) there exists a maximal knotless graph of size \(|E|\). We classify the maximal knotless graphs through nine vertices and 20 edges. We determine which of these maxnik graphs are the clique sum of smaller graphs and construct an infinite family of maxnik graphs that are not clique sums.
ISSN:2331-8422
DOI:10.48550/arxiv.2101.05241