History of Four Color Theorem

This is one of the most famous mathematical theories or problems. Many years were spent for finding the solution of this at least 150-year old problem. Mathematicians and some curious amateurs thought on this in different times. Some could not find any solution and some others, in their own words, did find the solution. Sometimes, a wrong solution was wrongly appreciated by many people for a long time. At time, the fault was identified and the correct solution was also provided.

What is the problem?

The easiest way to understand the problem is to color a map. Let us take the map of Bangladesh as an example.  It has 64 districts. If we want to color each side-by-side districts with different colors, how many colors will be needed? The Four Color Theorem says there will be maximum 4 colors needed. In some cases, may be 2 or 3 colors will be sufficient. The theory is not only about the map of Bangladesh; it is true for any maps of a country or place. In mathematical words, a plane surface divided into any number of blocks can be colored by blocks with maximum 4 different colors so that no two adjacent blocks will be of same color.


It is well accepted that the first formal question on this problem was raised by Francis Guthrie in 1852, that is, the question was about how many colors would be necessary and what was the proof with explanation. Guthrie was a student of University College, London at that time. He was then a student of famous Mathematician De Morgan who could understand the depth of the question. De Morgan wrote a letter and sent this problem to another milestone Mathematician Hamilton in Dublin. In reply Hamilton told it was not possible to solve in a short time. In the mean time, Francis Guthrie made some calculations on this what he gave to his younger brother Frederick Guthrie who was also a student of De Morgan. Later Guthrie became a lawyer in profession, but his passion and expertise in Mathematics destined him to a Mathematics Professor in his later age. During this period, De Morgan could not find the solution himself and continued searching for someone who might be able to solve this. Charles Pierce of USA worked for this up to the end of his life since 1860. In 1879 Mathematician Cayley understood the problem from De Morgan and prepared a research paper, for the Royal Geographical Society, on the difficulties of finding the solution of this problem. After a short while of this article published an declaration was published in the Nature journal where it was claimed that the solution for the Four Color Theorem was found. The solutions was made by a Barrister Alfred Bray Kempe who was a student of Cayley. With the advice of his teacher, Kempe sent the full solution to the American Journal of Mathematics. There another expart Story made some modifications in the article and sent to the Scientific Association of John’s Hopkins University. A number of renowned Mathematicians discussed on this article and Kempe got the recognition of his Kempe Chain Method. Kempe proved that the any map can be colored with maximum 4 colors for which he got awards from many prestigious sources including the Royal family of England. In 1880, Kempe published two more articles on the same problem as the improvements of his previous theories. These articles brought the famous Philosopher and Mathematician Professor P G Tait. He wrote at least two explanatory articles on this and established the problem as Four Color Theorem.

Until 1890, nobody could identify that the theories of Kepme and Tait were wrong and full of cleverly traits. Percy John Heawood described the error in Kempe’s Theorem for the first time. He claimed at that time 4 different colors will not be sufficient for all maps. Sometimes, may be 5 colors will be necessary. In 1896 Vallee Poussin identified the error in Kempe’s Theorem again without knowing the article of Heawood. Later, Kempe confessed his error by saying that at the time of writing that article it was not possible to identify and rectify the error of the Theory. Heawood was continuously working on the problem and showed the first Mathematical way for solving the problem. In later years, many Mathematicians from around the world worked and contributed for this. Every small contributions made some improvements in the theory. After a long journey, in 1976, the correct and complete Four Color Theorem was formed. It was developed jointly by Appel and Haken who were supported by computer works by Koch. In fact, it is one of the few theories that was established with huge computer supports. A significant branch in Mathematics Graph Theory was the base for establishing this theory where important theories of Euler and Hamilton were used. Other theories were also taken from Weinicke, Veblen, Franklin, Birkhoff and Heesch. Actually, Appel and Haken used the Kempe Chain Method. They published their works in two sequels in 1977. Since then, many people worked for improvements of the theory and eventually made this simple. In 2000, Ashay Dharwadker made a complete different proof of the theorem and he is awarded by Canadian Mathematical Society and other organizations.

Please read the next part of this article Graph Theory as the Base of Four Color Theorem.

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