Graph Theory: The Four Coloring Theorem

Graph Theory: The Four Coloring Theorem

"Every planar map is four colorable," seems like a pretty basic and easily provable statement. However, this simple concept took over one hundred years and involved more than a dozen mathematicians to finally prove it. Throughout the century that many men pondered this idea, many other problems, solutions, and mathematical concepts were created. I find the Four Coloring Theorem to be very interesting because of it's apparent simplicity paired with it's long, laborious struggle to be proved. There is a very long and eventful history that accompanies this theorem.

The concept of the Four Coloring Theorem was born in 1852 when Francis Guthrie noticed that he only needed four different colors to color in a map of England. Through his brother, Frederick, Francis communicated his discovery to De Morgan. Francis wondered if De Morgan would be able to tell him if it was true or not. De Morgan was unsure, so he asked the same question to Hamilton in Dublin. Hamilton was unable to help, so De Morgan continued to ask other prominent mathematicians. In the US, Charles Peirce attempted to prove the Four Color Conjecture in the 1860's and continued to for the remainder of his life. In 1879, Cayley wrote a paper to the Royal Geographical Society explaining the difficulties in attempting to prove the Conjecture. On July 17, 1879, a mathematician by the name of Kempe announced a proof for the Four Color Conjecture. However, eleven years later Heawood, a lecturer at Durham England, pointed out that Kempe's proof was incorrect. Along with proving Kempe wrong, Heawood was able to prove that every planar map is five colorable. In 1898, Heawood also proved that if the number of edges around a region is...

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...actually quite fun as well. They don't really have a real importance in the real world. The Four Color Theorem isn't going to save any lives or make life that much easier. However, it does make map coloring more simple by requiring only four colors.

Bibliography

(1) Fritsh, Rudolf and Gerda, The Four-Color Theorem, Springer-Verlag, New York, Inc., 1998.

(2) Harary, Frank, Graph Theory, Adison-Wesley Publishing Co., Redding, MA, 1972, p.130-131.

(3) Kainen, Paul, and Saaty, Thomas, The Four Color Problem, McGraw-Hill, Inc., Great Britain, 1977.

(4) The Four Color Theorem, http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.html, December 10, 1999.

(5) The Four Color Theorem, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas, http://www.math.gatech.edu/~thomas/FC/fourcolor.html, December 10, 1999.