Theorem:

Every regular map is four colorable. For any subdivision of the plane into non-overlapping regions, it is always possible to mark the regions with four colors, in such a way that no two adjacent regions receive the same color.

Outline of the proof:

Instead of considering maps that can be really different in shape one from the other, this proof simplifies and transforms all maps into a common and more manageable form. This way it will be easier to handle the complexity behind all possible shapes that maps can have. The proof is split into these parts:

• All maps can be simplified by removing all faces with less than five edges, without affecting the search and the validity of the proof (see T1). 02/Apr/2011 – This theorem was already known. It was found by Kempe back in 1879 in terms of graph theory (see http://en.wikipedia.org/wiki/Four_color_theorem: “Kempe also showed correctly that G can have no vertex of degree 4″).
• All maps can be topologically transformed into circular maps or rectangular maps to make it easier to handle them (see T2)
• Core of the proof

Core of the proof:

Sooner or later … or much later I will find a solution. For now, read the posts and the “Open points and notes“.

And even if I’ll never find it, it is fun to try!