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.