https://blogs.scientificamerican.com/roots-of-unity/4-color-map-theorem/

This article is talking about the 4-color theorem and coloring maps for countries. It says that when you’re coloring a map with states or areas that are not connected (such as the US—Hawaii and Alaska) that this theorem does not necessarily work because everything is not connected on one map. Africa is considered a 4-colorable map, while South Africa does not work with this theorem.

This article also talks about the difference between a computer testing the theorems and humans testing them. It is obvious that a computer will most likely be able to come up with the most efficient coloring pattern, while a human may not initially do so. It may take a human a number of times to come up with the optimal solution for coloring with the least amount of colors.

Interestingly enough, this is something I was actually kind of thinking about as the four color theorem was being introduced to us in class. I was able to figure out that the four color theorem may not hold if two or more locations are bound so that they must share the same color.

Professor Plante sort of demonstrated this and what is brought forward in your article, as he introduced the coloring of 3D models using 2D projections. As he showed, 3D models, such as a Möbius band or a donut, can be colored, using 2D projections while bypassing the four color theorem. The catch is, that there are restrictions to the 2D projections that link certain parts of the models. Notice anything similar? Because of these restrictions we are able to bypass the four color theorem. So in the case of regions linked together, such as that of Hawaii or Alaska, it may be useful to bypass the four color theorem and also think of these maps as 2D projections of 3D models.