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G=(v,e) is a simple planar graph with |v|>10 vertices. I need to prove that G#=(V,E#)-the complement of G- is not a plannar graph.

I tried to use Euler's formuala, but it didnt went well.

Roey Waitsman
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1 Answers1

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Suppose that the total number of vertices of $G$ is $n$ and $n \geq 11$. By the Euler formula, we know that if $G$ is a planar graph, then $|E(G)| \leq 3n-6 $. On the other hand, $E(G)+E(\bar G) =\binom{n}{2}.$ So $E(\bar G) = \binom{n}{2} -E(G) \geq \frac{n(n-1)}{2}-3n+6 > 3n -6$ (since $n \geq 11$) and so $\bar G$ is not a planar graph.

Ken
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