The Four-Colour-Problem

Yesterday I was a bit bored, and then I recalled having read the biography of a German mathematician called Heinrich Heesch several years ago, who spent most of his life on solving the so-called four-colour problem. (http://en.wikipedia.org/wiki/Four_color_theorem) The four-colour problem is about the question whether four colours are always enough to colourize a two-dimensional map in such a way that there are no adjacent countries having the same colour.

I pondered over this problem and after some time I came to the conclusion that the problem is trivial: Every two-dimensional map can be represented by a planar graph that has no crossings. (I called it a "two-dimensional graph".) In such a graph, it is possible to draw up to 4 nodes which have edges leading to each other, but it is not possible to draw more than 4 - otherwise, there would be a crossing. Therefore, four colours always suffice.

The "official" proofs of this problem, which is now called a theorem, are very complicated and consider more than 1000 special cases - I wonder why? Have I overseen something, or have the professional mathematicians not noticed that it's not possible to draw more than 4 nodes having edges leading to each other in a planar graph without crossings? For example, the proof at http://people.math.gatech.edu/~thomas/FC/fourcolor.html contains special cases in which there are crossings. Due to my argumentation, it becomes clear that these special cases are not relevant to the problem!

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