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Proposition: A Necessary Condition for a Graph to be Planar
Let $G(V,E)$ be a simple, biconnected, and planar graph. Then the number of its edges is less than three times the number of its vertices less $6$, formally $$|E|\le 3|V|-6.$$
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- Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000