Although it may be surprising, there is actually a whole topic in graph theory that relates to the 4-color graph theorem. The 4-color graph theorem was proven. Then this proof was found invalid, and then was proven again. The actual proof is a 100 pages long and contains a lot of similarities to 5-color and 6-color proofs.

Here are some terms that you would need to know:

Proper coloring: A coloring in which now to colors share any adjacent sides or vertices (or no colors are touching).
Planar graphs: A planar graph is a graph that can be drawn in such a way that no edges cross.
              Here are some examples of Planar Graphs: