For the four coloring problem, maps can be pre-simplified removing all faces that have 2, 3 or 4 edges. In this case the Euler identity becomes: $F_5 + 0F_6 = 12 + F_7 + 2F_8 + 3F_9 + …$. One thing that can be noticed is that pre-simplified maps must have a great number of faces of type $F_5$ (with 5 edges) respect to faces of type $F_7$ or greater. For complex maps (containing faces with a lot of edges) many $F_5$ have to exist. For example for a single face with 1000 edges, 994 faces of type F5 must be added to the Euler identity ($F_5 = … + 994 F_{1000} + …$).
One thing that I initially imagined is that these $F_5$ faces were grouped in cluster, in which an $F_5$ face is surrounded only by other $F_5$ faces. Then I realized that $F_5$ faces could also be mixed with faces of type $F_6$ (which do not alter the Euler identity) or with faces of type >= $F_7$.
I was looking for a paper that analyze the distribution of faces in maps (possibly pre-simplified). Can anybody help?
Just for sharing (not related to this question), one thing that I found analyzing this is that simplified maps of 13 faces do not exist.