Is there an English translation of Kuratowski's proof about planar graphs?
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$\begingroup$ You want a translation of his proof, or a proof? $\endgroup$– Mariano Suárez-ÁlvarezCommented Dec 24, 2009 at 21:21
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$\begingroup$ I am interested in the translation (mostly). $\endgroup$– adamoCommented Dec 25, 2009 at 18:55
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3 Answers
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In case you are asking for the original paper "Sur le problème des courbes gauches en Topologie" by Kuratowski where he first proves his characterization of planar graphs, then a translation by J.Jaworowski can be found in "Graph Theory, Łagów", 1981, M. Borowiecki, J. W. Kennedy and M. M. Sysło. It is the proceedings of a conference held in Łagów, dedicated to the memory of K.Kuratowski.
answered Dec 24, 2009 at 21:57
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2$\begingroup$ Here are links to the French original and to the English translation (by Jaworowski) published in those proceedings. springerlink.com/content/7021318369w65376 matwbn.icm.edu.pl/ksiazki/fm/fm15/fm15126.pdf $\endgroup$ Commented Dec 25, 2009 at 3:49
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Kuratowski's theorem is set as an exercise in Ch. 5 of Combinatorial Problems and Exercises by Lovasz. The problem is given as follows:
Let G be a minimal non-planar graph with all vertices of degree at least 3. Then:
- G is 3-connected. (This is straightforward; supposing otherwise and removing the cutset we can get a planar embedding of G.)
- G contains a cycle with a chord. (Provided hint: Consider a maximum path.)
- G is isomorphic to one of $K_5$, $K_{3, 3}$.
- Conclude Kuratowski's theorem from part 3.
The proof is on pp. 299-301, which fortunately are all viewable in the Google Books preview.
answered Dec 24, 2009 at 22:54
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You might just want to read Jim Belk's excellent exposition at the Everything Seminar.
answered Dec 24, 2009 at 22:08
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$\begingroup$ I believe his name is Jim Belk. $\endgroup$ Commented May 4, 2010 at 14:06