# Kuratowski’s theorem

A finite graph is planar if and only if it contains no subgraph that is isomorphic to or is a subdivision of $K_{5}$ or $K_{3,3}$, where $K_{5}$ is the complete graph of order 5 and $K_{3,3}$ is the complete bipartite graph with 3 vertices in each of the halfs. Wagner’s theorem is an equivalent later result.

## References

• 1 Kazimierz Kuratowski. Sur le problème des courbes gauches en topologie. Fund. Math., 15:271–283, 1930.
Title Kuratowski’s theorem KuratowskisTheorem 2013-03-22 11:57:45 2013-03-22 11:57:45 bbukh (348) bbukh (348) 12 bbukh (348) Theorem msc 05C10 PlanarGraph WagnersTheorem