 # The Mathematics of Map Coloring

explanatory Essay
1881 words
1881 words The Mathematics of Map Coloring

The four-color conjecture has been one of several unsolved mathematical problems. From 1852 to this day, practically every mathematician has studied the problem long and hard, but to no avail. The conjecture looks as though it has been solved by Wolfgang Haken and Kenneth Appel, both of the University of Illinois. They have used computer technology to prove the conjecture. The calculation itself goes on for about 1200 hours. The staggering length of the computation of the proof is what creates some controversy in the mathematical world. The Appel-Haken Theorem is based on numerous assumptions, “that there is an overwhelmingly great probability that their method of proof must succeed.”  It assumes that the theory itself is correct, but the theory itself is also an assumption. You can see why this issue has been wreaking havoc for many years.

It all started back in 1852 when Francis Guthrie was coloring a map of England. He wanted to know the least amount of colors, or chromatic number, it would take to color the map so no two adjacent regions are of the same color. He found the chromatic number to be four. He then studied arbitrary maps and wondered if all maps could be colored with four colors. Francis’ curiosity would be in the minds of all mathematicians to come. He then passed this question on to his brother, Frederick. He then submitted this to his professor Augustus deMorgan as a mathematical conjecture.

deMorgan was fascinated by the Four-Color problem and wrote in a letter to his colleague Sir William Rowan Hamilton the next day after seeing the conjecture. Hamilton was less enlightened by it, and never worked on it. It was through deMorgan that the Four-Color problem was made known, thus deMorgan has incorrectly been dubbed the originator of the problem.

Eventually the hype surrounding the conjecture died down in the early 1860’s. This down time, during which interest in the problem was minimal, only lasted about twenty years. A lawyer by the name of Alfred Bray Kempe proposed a solution in The American Journal of

#### In this essay, the author

• Explains how to delete the vertex and all edges leading into it until 6 vertices remain.
• Explains how to color the vertex with a color that isn't used on adjacent vertices.
• Introduces rudolf and gerda pritsch's the four color theorem.
• Explains that the four-color conjecture has been one of several unsolved mathematical problems.
• Assumes the opposite of the lemma, that no vertex on the graph has order five or less.
• Explains that the proof of the five-color theorem is not necessary since it has very similar characteristics to the six-colored proof.
• Explains that coloring a map is important because it can be applied to areas outside of cartography.