Question

Is there an English translation of Kuratowski's proof about planar graphs?

Answer 1

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.

Answer 2

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:

1. G is 3-connected. (This is straightforward; supposing otherwise and removing the cutset we can get a planar embedding of G.)
2. G contains a cycle with a chord. (Provided hint: Consider a maximum path.)
3. G is isomorphic to one of $K_5$, $K_{3, 3}$.
4. Conclude Kuratowski's theorem from part 3.

The proof is on pp. 299-301, which fortunately are all viewable in the Google Books preview.

Answer 3

You might just want to read Jim Belk's excellent exposition at the Everything Seminar.