Four color theorem and map pre-simplification of faces with less than 5 edges It is already known that in searching for a solution of the four color problem, regular maps can be pre-simplified by removing all faces with less than four edges. This is described for example in the book "What is Mathematics? An Elementary Approach to Ideas and Methods" about the five color theorem.

I belive that all regular maps can be simplified by removing all faces with less than five edges (instead of less then four), without affecting the search and the validity of the four color theorem. This simplification is described here: http://4coloring.wordpress.com/t1/

In this case Euler’s identity gets really simplified: F5 = 12 + F7 + 2F8 + 3 F9 + ...

What is known about this? Has it already been studied before?
The result is known since Kempe in 1879, and is mentioned in http://en.wikipedia.org/wiki/Four_color_theorem as "Kempe also showed correctly that G can have no vertex of degree 4...."  Your proof does not work, because when you remove two edges joining face B with A and C, A and C may already have been adjacent.  Since you give the combined face a color, then return face B with a new color, you leave A and C adjacent and of the same color.