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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 image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier