A Triangulated Planar Map is created from any planar map in two steps;
1. The faces are made into polygons by straightening each border.
2. The map is triangulated by triangulating each face polygon.
AXIOM: A T-map; ie, a fully triangulated planar map, is always 4 face colorable.
For the skeptical: Every face has only three neighbors. Therefore, at least
one of the four colors will be available to color a face.
The graph dual of a T-map is a cubic planar graph; ie CPG. The edge colorability
of a CPG is immaterial; since the generating T-map is inevitably 4 face colorable.
The graph dual of a 3-regular map is a fully triangulated planar graph; sometimes
called a "triangulation". The 4CT is true if and only if every graph triangulation
is 4 vertex colorable. The 4 face colorability of a g-t does NOT prove the 4CT.