The Four Color Theorem states that every planar graph can be colored using no more than four colors in such a way that no two adjacent vertices have the same color. This theorem was first proposed in 1852 by Francis Guthrie, and it was finally proved in 1976 by Kenneth Appel and Wolfgang Haken. However, their proof relies heavily on computer assistance and is not suitable for presentation in a simple text-based format.
The proof by Appel and Haken involved the concept of reducibility and the idea that certain configurations in a planar graph can be simplified without changing the graph's colorability. They identified a set of 1,936 "unavoidable" configurations and showed that each one is "reducible." The proof required checking each of these configurations with the aid of a computer.
To provide a full proof of the Four Color Theorem here would be impractical due to its complexity and reliance on computer-verified components. Nevertheless, I can give a general outline of the approach:
Suppose there exists a minimal counterexample to the Four Color Theorem, i.e., a planar graph G with the smallest number of vertices that cannot be properly colored using only four colors.
By applying Euler's formula for planar graphs (V - E + F = 2), it can be shown that the graph G must contain a vertex of degree 5 or less.
From here, several cases are considered depending on the degree of the vertex:
a. If G has a vertex of degree 3 or less, then removing the vertex simplifies the graph, and a proper 4-coloring for the simplified graph can be extended to the original graph, contradicting the assumption that G is a minimal counterexample.
b. If G has a vertex of degree 4, further analysis can be done, either finding a contradiction or showing that a specific configuration is reducible.
c. If G has a vertex of degree 5, a similar approach as in case b is taken, analyzing the configurations surrounding the vertex.
The analysis of the degree 4 and 5 cases requires considering a large number of subcases and configurations. Appel and Haken identified 1,936 unavoidable configurations that must be present in a minimal counterexample, if it exists.
Using computer assistance, they checked each of the 1,936 configurations and showed that they are all reducible. Thus, any minimal counterexample must contain a reducible configuration, which leads to a contradiction.
Therefore, the Four Color Theorem holds true for all planar graphs.
Keep in mind that this is only an outline of the proof, and the actual proof is much more involved and requires extensive case analysis and computer assistance.