The four color theorem appeared in 1852, talking about the problem of coloring real maps. Let’s examine some basic aspects of these maps in relation to the four color theorem.

A world with just water and one land with no divisions, topologically equivalent to a disk, needs only two colors to paint the land and the ocean. This is the beginning, as the Pangea on Earth long ago.

If two parties want to share this land and both want to have access to the ocean and have contiguous regions, there is only one possible configuration, and the resulting map can be represented by a planar graph with 2 vertexes and 3 edges (multiedge graph). The other solutions that have been excluded by the two restrictions: “have access to the ocean” and “contiguous regions” are not so interesting for the scope of the theorem (see previous post).

If a third party comes into play, the initial region (surrounded by the ocean) has to be split among three parties, with the same two restrictions as stated before: “have access to the ocean” and “be contiguous regions”. If we consider all possible combinations, we can also see that among them there are some that can be eliminated introducing a third rule: “all faces have to touch each other”. This third rule can be added because all the configurations in which the three faces don’t touch each other contain F2 faces and therefore (is it to be proved?) can be eliminated. It will then rest only one configuration that meet all criteria, that is the one printed at the bottom right of the previous post image.