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?