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# 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*

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new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag