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Determine exactly the values of $m$ and $n$ for which the complete bipartite graph $K_{m,n}$ is planar.

I have tried doing this by drawing different complete bipartite graphs and just using guess and check to see if planar or not. Obviously this isn't working and would like to see how this is done.

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migrated from stats.stackexchange.com May 29 '15 at 3:40

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Knowing that $K_{3,3}$ is non-planar helps. Also, by Kuratowski's theorem any graph that has a subgraph homeomorphic to $K_{3,3}$ is also non-planar. This should narrow it down quite a bit because we can now conclude that $K_{3,n}$ and $K_{m,3}$ are non-planar for $m,n \geq 3$. Can you take it from here?

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