The four-color conjecture was put forward by Britain. 1852, when Francis guthrie, who graduated from London University, came to a scientific research unit to do map coloring, he found an interesting phenomenon: "It seems that every map can be colored with four colors, so countries with the same border will be colored with different colors." Can this phenomenon be strictly proved by mathematical methods? He and his younger brother, Grace, who is in college, are determined to give it a try. The manuscript papers used by the two brothers to prove this problem have been piled up, but the research work has not progressed. 1852, 10 year123 October, his younger brother asked his teacher, the famous mathematician de Morgan, for proof of this problem. Morgan couldn't find a solution to this problem either, so he wrote to his good friend, Sir Hamilton, a famous mathematician, for advice. Hamilton demonstrated the four-color problem after receiving Morgan's letter. But until the death of 1865 Hamilton, this problem was not solved. 1872, Kelly, the most famous mathematician in Britain at that time, formally put forward this question to the London Mathematical Society, so the four-color conjecture became a concern of the world mathematical community. Many first-class mathematicians in the world have participated in the great battle of four-color conjecture. During the two years from 1878 to 1880, Kemp and Taylor, two famous lawyers and mathematicians, respectively submitted papers to prove the four-color conjecture and announced that they had proved the four-color theorem. Everyone thought that the four-color conjecture was solved from now on. Kemp's proof is as follows: First of all, it is pointed out that if no country surrounds other countries, or no more than three countries intersect at one point, the map is called "regular". A map is often linked by a regular map and an informal map, but the number of colors required by an informal map generally does not exceed the number required by a regular map. If a map needs five colors, it means that its regular map is five colors. To prove the four-color conjecture, it is enough to prove that there is no regular five-color map. Kemp proved this point by reducing to absurdity, to the effect that if there is a regular five-color map, there will be a "minimal regular five-color map" with the least number of countries. If there is a country with less than six neighboring countries in the minimal regular five-color graph, then there will be a regular graph with fewer countries that is still five-colored, so there will be no country with minimal five-color graph, and there will be no regular five-color graph. So Kemp thought he had proved the "four-color problem", but later people found him wrong. However, Kemp's proof clarifies two important concepts and provides a train of thought for solving problems in the future. The first concept is "configuration". He proved that in every regular graph, at least one country has two, three, four or five neighbors, and there is no regular graph in which every country has six or more neighbors. In other words, a set of "configurations" consisting of two neighboring countries, three neighboring countries, four or five neighboring countries is inevitable, and each map contains at least one of these four configurations. Another concept put forward by Kemp is reducibility. The use of the word "negotiable" comes from Kemp's argument. He proved that as long as there is a country with four neighboring countries in the five-color map, there will be a country with fewer five-color maps. Since the concepts of "configuration" and "reducibility" were put forward, some standard methods for testing configurations to determine whether they are reducible have been gradually developed, and the inevitable groups of reducible configurations can be found, which is an important basis for proving the "four-color problem". However, to prove that a large configuration is negotiable, many details need to be checked, which is quite complicated. 1 1 years later, that is, 1890, only 29-year-old, studying at Oxford University, pointed out the loopholes in Kemp's certificate with his own accurate calculation. He pointed out that Kemp's reason that a country without a minimum five-color map cannot have five neighboring countries is flawed. Soon, Taylor's proof was also denied. People found that they actually proved a weak proposition-the five-color theorem. In other words, it is enough to paint the map with five colors. Later, more and more mathematicians racked their brains for this, but found nothing. As a result, people began to realize that this seemingly simple topic is actually a difficult problem comparable to Fermat's conjecture. Since the 20th century, scientists have basically proved the four-color conjecture according to Kemp's idea. 19 13. boekhoff, a famous American mathematician and Harvard University, used Kemp's ideas and combined his new ideas. It is proved that some large configurations are reducible. Later, American mathematician Franklin proved in 1939 that maps below 22 countries can be colored in four colors. 1950 someone has been promoted from 22 countries to 35 countries. 1960 proves that maps below 39 countries can be colored with only four colors; And then push it to 50 countries. It seems that this progress is still very slow. The invention of high-speed digital computer urges more mathematicians to study the "four-color problem". Heck, who began to study the four-color conjecture from 1936, publicly declared that the four-color conjecture can be proved by finding the necessary group of reducible graphs. His student Toure wrote a calculation program. Heck can not only prove the reducibility of the configuration with the data generated by this program, but also describe the reducible configuration by transforming the mapping into a shape called "duality" in mathematics. He marked the capital of each country, and then connected the capitals of neighboring countries with a railway crossing the border. Except for the capital (called vertex) and the railway (called arc or edge), all other lines have been erased, and the rest are called dual graphs of the original graph. In the late 1960s, Heck introduced a method similar to moving charges in an electrical network to find an inevitable set of configurations. The "discharge method", which first appeared in a rather immature form in Heck's research, is a key to the future study of inevitable groups and a central element to prove the four-color theorem. After the emergence of electronic computers, the process of proving the four-color conjecture has been greatly accelerated due to the rapid improvement of calculation speed and the emergence of man-machine dialogue. Harken of the University of Illinois began to improve the "discharge process" in 1970, and then compiled a good program with Appel. 1June, 976, they spent 1200 hours on two different electronic computers of the University of Illinois in the United States, made 1000 billion judgments, and finally completed the proof of the four-color theorem, which caused a sensation in the world. This is a great event that 100 has attracted many mathematicians and math lovers for more than a year. When two mathematicians published their research results, the local post office stamped all the mails sent that day with a special postmark of "four colors are enough" to celebrate the solution of this problem. The "four-color problem" proved to be only a solution to a difficult problem that lasted for more than 100 years, and it became the starting point of a series of new ideas in the history of mathematics. In the research process of "four-color problem", many new mathematical theories have emerged and many mathematical calculation skills have been developed. For example, turning the coloring problem of maps into a graph theory problem enriches the content of graph theory. Moreover, the "four-color problem" has also played a role in effectively designing airline flight schedules and designing computer coding programs. However, many mathematicians are not satisfied with the achievements made by computers. They think there should be a simple and clear written proof method. Today, many mathematicians and math lovers are still looking for a more concise proof method.

Satisfied, please adopt.