Five-Color Theorem - Essay Example

It all began with Francis Guthrie. He was a mathematician from British, in 1952 discovered that he could color the states in the map of Great Britain using four colors without coloring 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 for the four-color problem and went ahead to publish his solution and proof. In 1890; however, P. Heawood discovered an error in Kemper's 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 to the color problem with the five-color theorem sufficing (Jensen and Toft 61).

To prove the five-color theorem mathematically, one relates a planar graph, G to a certain map. A vertex is placed on every area on 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 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 planar. This is to denote that G may be embedded in a plane without necessarily intersecting the borders. Now take out the vertex V, from the planar graph, G. The new graph attained after this will have one vertex less than the original graph G.

At this point, we can presume, by induction that this graph can be painted with just five colors. The vertex V must be linked with five other vertices because if not, it can be painted with the planar graph, G, with a paint not used by the other vertices. Look at the five vertices V1, V2, V3, V4, and V5 that were adjacent to V in cyclical order. It will be found that less than five colors in painting these vertices; therefore, the vertex V can be painted to render the planar graph with five coloreds. At this point, we assume that V1, V2, V3, V4, and V5 are colored with colors 1, 2, 3, 4, and 5 respectively. Now consider the sub-graph G13 of the reduced G which consists of the vertices colored with colors 1 and 3 only and the points that connect both. If V1 and V3 occur in separate linked components, in G13, the coloration can be reversed to that having V1. It is then possible to give the color 1 to V hence solving the problem (Jensen and Toft 62).

If on the contrary, v1 and v3 are appearing in the same region on G13 it is possible to join them. A path is created joining the points that are colored with color one and color three. Turn to the subgraph G24 of the reduced G. This subgraph contains the points that have either color two or color four.  Use arguments in the first subgraph G13 as above. It is now either realistic to reverse the coloration on a sub-graph of G24 and paint V with color 2, alternatively, we can join the point V2 with the vertex V4 through all the points that contain their respective colors. It is; therefore, clear that this path through G24 would cut through the path established in the sub-graph G13. We can conclude that G can be colored using five colors, which is the complete opposite of the first assumption. This proves the latter, that the map can be colored using at least five colors, which essentially is the five-color theorem (Jensen and Toft 62).