## You are here

HomeKuratowski's theorem

## Primary tabs

# 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.

Keywords:

planar

Related:

PlanarGraph, WagnersTheorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

05C10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Sep 17

new question: numerical method (implicit) for nonlinear pde by roozbe

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella

new question: numerical method (implicit) for nonlinear pde by roozbe

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella