# 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 |
---|---|

Canonical name | KuratowskisTheorem |

Date of creation | 2013-03-22 11:57:45 |

Last modified on | 2013-03-22 11:57:45 |

Owner | bbukh (348) |

Last modified by | bbukh (348) |

Numerical id | 12 |

Author | bbukh (348) |

Entry type | Theorem |

Classification | msc 05C10 |

Related topic | PlanarGraph |

Related topic | WagnersTheorem |