| Version current |
Version 7 |
| 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. |
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 of order 6. Wagner's theorem is an equivalent later result. |
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| \begin{thebibliography}{1} |
\begin{thebibliography}{1} |
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| \bibitem{cite:kuratowski_planarity} |
\bibitem{cite:kuratowski_planarity} |
| Kazimierz Kuratowski. |
Kazimierz Kuratowski. |
| \newblock Sur le probl{\`e}me des courbes gauches en topologie. |
\newblock Sur le probl{\`e}me des courbes gauches en topologie. |
| \newblock {\em Fund. Math.}, 15:271--283, 1930. |
\newblock {\em Fund. Math.}, 15:271--283, 1930. |
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| \end{thebibliography} |
\end{thebibliography} |
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| %@ARTICLE{cite:kuratowski_planarity, |
%@ARTICLE{cite:kuratowski_planarity, |
| % author = {Kazimierz Kuratowski}, |
% author = {Kazimierz Kuratowski}, |
| % title = "Sur le Probl{\`e}me des Courbes Gauches en Topologie", |
% title = "Sur le Probl{\`e}me des Courbes Gauches en Topologie", |
| % journal = {Fund. Math.}, |
% journal = {Fund. Math.}, |
| % volume = 15, |
% volume = 15, |
| % pages = {271--283}, |
% pages = {271--283}, |
| % year = 1930 |
% year = 1930 |
| %} |
%} |
