Curiously, the theorem implies all maps on a sphere can also be colored. All 2D maps can be considered to be a sphere, by adding a new 'border state' that surrounds the existing 2D map and extends endlessly in all directions. The 2D theorem says this can also be colored.
That can be mapped to a sphere by laying the existing map on a sphere, and making that border state into the 'rest of the sphere'. You have to stretch things, sure. The colorability of the map is not affected by resizing or stretching the map. The fundamental layout of the map is determined by the vertices and edges, not how you 'look at it'.
This also implies that all 2D maps must be colorable in 3 colors on their perimeter. Because, I can always create a new map with that 'border state' that must also be colorable, and the new state can't be any color already on the perimeter. So, the perimeter can only be 3 colors.
The spherical transformation makes this more obvious: every state must have neighbors of at most 3 colors, and that border state is no different from any other.
