Hi, need to submit a 500 words essay on the topic 5-color theorem.There are three of them, four-color, five-color and six-color theorem. The five color theorem was proved in 1890 showing that five col

Hi, need to submit a 500 words essay on the topic 5-color theorem.

There are three of them, four-color, five-color and six-color theorem. The five color theorem was proved in 1890 showing that five colors suffice to color a map. (Jensen and Toft 61)

It all began with Francis Guthrie. He was a mathematician from British, who in 1952 discovered that he could color the states in the map of Great Britain by means of four colors without coloring of the neighboring countries with the same color. The problem hence arose if it was feasible to color any given map using four colors and it remained an area of interest for a while. The problem was. however, deciphered in 1879 when A. Kempe claimed to have found an explanation to the four color problem and went ahead to publish his solution and proof. In 1890. however, P. Heawood discovered an error in Kempers proof, which led to the demotion of the four color theorem as a credible theory. Heawood was unable to show that there was an error, which could have been colored with not less than five colors, but ultimately proved that Kempe was wrong in his argument. This led to a solution in the color problem with the five color theorem sufficing (Jensen and Toft 61).

In order to proof the five color theorem mathematically, one relates a planar graph, G to a certain map. A vertex is placed on every area in the map. Two vertices are then connected with an edge where analogous areas share a boundary in common. This problem is then translated into a graph coloring problem. One is now required to color the graph vertices so that no border has its endpoints with a similar color. This proof relies heavily on the Euler characteristic to illustrate that there, it is mandatory to have a vertex V that is shared by at most five borders. It also relies on the fact that G is a planar. This is to denote that G may be embedded in a plane without necessarily intersecting the borders. Now take out