(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that almost all graphs are not planar graphs.

2. Relevant equations

Kuratowski's Theorem

3. The attempt at a solution

Kuratowski's Theorem states that a graph is planar if and only if it does not contain a subgraph homeomorphic toK_{5}orK_{3,3}. Therefore, if I can show almost all graphs have subgraphs homeomorphic toK_{5}orK_{3,3}, this would complete the proof.

My thought on this was to letP_{n}denote the probability that a random graph onnvertices is not planar and then show lim_{n -> ∞}P_{n}= 1. However, I need some help on how to show which graphs onnvertices contain subgraphs homeomorphic toK_{5}orK_{3,3}. Can anyone give me a nudge in the right direction?

Thanks!

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# Almost All Graphs are Non-Planar

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