Four color theorem

The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point.

This theorem was conjectured in 1853 by Francis Guthrie[?]. It is obvious that three colors are completely inadequate, and mathematicians were able to prove that five colors were sufficient to color a map. However, it was not until 1977 that the conjecture was finally proven by Ken Appel[?] and Wolfgang Haken[?]. The proof reduced the infinitude of possible maps to 1936 configurations (later reduced to 1476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson[?], Daniel Sanders[?], Paul Seymour[?] and Robin Thomas[?] produced a similar proof which required to check 633 special cases. This new proof also does contain parts which require the use of a computer and can't be checked by humans alone.

The four color theorem was the first major theorem to be proven using a computer, and the proof was not accepted by all mathematicians because it could not directly be verified by a human. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See experimental mathematics.

The lack of mathematical elegance was another factor; to paraphrase comments of the time "a good mathematical proof is like a poem -- this is a telephone directory!"

Formal statement in graph theory

To formally state the theorem, it is easiest to rephrase it in graph theory. It then states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color. Or "every planar graph is four-colorable" for short. Here, every region of the map is replaced by a vertex of the graph, and two vertices are connected by an edge if and only if the two regions share a border segment.

Generalizations

One can also consider the coloring problem on surfaces other than the plane. The problem on the sphere is equivalent to that on the plane. For closed (orientable or non-orientable) surfaces with positive genus[?], the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

<math>p=[(7 + \sqrt{(49 - 24 \chi)}) / 2]</math>

where the brackets denote the floor function. The only exception to the formula is the Klein bottle, which has Euler characteristic 0 and requires 6 colors. This was initially known as the Heawood conjecture and proved by Gerhard Ringel[?] and J. T. W. Youngs[?] in 1968.

Real world counter examples

In the real world, not all countries are contiguous (e.g. Alaska as part of the US), and one typically requires that all parts of a country receive the same color. Under these conditions, the corresponding graph need no longer be planar, and four colors are not always sufficient. Consider for instance the map

                  -----------
                 |     |     |
                 | D   |  C  |
               --|   ----    |
              |  |  |    |   |
              |   --| A  |   |
          ----|  |   ----    |
         |    |  |  B   |    |
         | A  |   ------ ----
          ----|    E       |
               ------------

where the two regions labeled 'A' belong to the same country. Five colors are required if those two regions are to receive the same color. Similarly, one can construct maps that require an arbitrary high number of colors.

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References:

• Appel, K. and Haken, W., Every Planar Map is Four-Colorable. Providence, RI: American Mathematical Society, 1989
• O'Connor and Robertson, History of the Four Color Theorem, MacTutor project, http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem