How to prove the four color theorem, the four color theorem and the ten color theorem

One of the three major mathematical problems in modern times. The four-color conjecture came from the United Kingdom. But it doesn't seem to have been proven yet.

The four-color theorem (one of the three major mathematical problems in modern times), also known as the four-color conjecture and the four-color problem, is one of the three major mathematical conjectures in the world.

The four-color question reads, "Any map with just four colors can paint a country with a common border in different colors." In other words, without causing confusion, a map only needs to be marked with four colors.

In mathematical terms, it is "arbitrarily subdivided into non-overlapping regions, each of which can always be marked with one of the four numbers 1234, without giving the same number to two adjacent regions." "By adjacent area, we mean that there is an entire section of boundary that is public. If two regions meet only one point or a finite number of points, they are not called adjacent.

The ten-color determination is also known as the Heawood theorem. In the process of trying to prove the four-color fixed height envy theory, human beings found that it was easier to construct a plane with 10 regions connected in pairs on a curved surface.

The four-color problem, also known as the four-color conjecture and the four-color theorem, is one of the three major mathematical problems in the modern world. (i.e., "Arbitrarily subdivide a plane into non-overlapping areas, each of which can always be marked with one of the four numbers 1234 without giving two adjacent regions the same number.") This is an unsolved mystery of the world's mathematics that has not been solved until now.

The four-color theorem was once solved by computers, but it was not recognized by mathematicians. So the four-color theorem has not been solved until now. However, in this process, there is a more famous Kemp, but it didn't take long for Herwood to deny it, and he proposed the five-color theorem on this basis, and the wide silver but Kemp's is not useless, and Kemp used the reductive method to solve it.

But Kempkey's stupid proof illuminates two important concepts that provide a way to solve the problem later. The first concept is "configuration". He proved that in every regular map there is at least one country with two, three, four, or five neighbors, and that there is no regular map in which every country has six or more neighbors, that is, a set of "configurations" consisting of two, three, four, or five neighbors is inevitable, and that each map contains at least one of these four configurations.

Another concept proposed by Kemp is "negotiability". The use of the word "renewable" comes from Kemp's argument. He proved that as long as one country has four neighbors in a five-color map, there will be a five-color map with fewer countries.

Since the introduction of the concepts of "configuration" and "reducibility", some standard methods for examining configurations to determine whether they are reducible have been gradually developed, and the inevitable group of reducible configurations can be sought, which is an important basis for proving the "four-color problem". But to prove that a large configuration is reducible, a large number of details need to be checked, which is quite complicated.

So after that, the four-color theorem began to develop slowly.

But when electronics came out, there was a solution to the four-color theorem. In June 1976, two different computers at the University of Illinois in the United States spent 1,200 hours and 10 billion judgments, and the result was that none of the maps needed five colors, and finally proved the four-color theorem. This news shocked the world, but this has not yet been recognized by the world, because it is calculated by computer.

So no one has yet been able to solve this with their own hands.

According to this question, several more questions arise. 1. The general coloring methods of point coloring, edge coloring, surface coloring, body melting coloring and the coloring method of non-plane drawings. These people will be quicker to apply computer programs using the workchart method.

To this end, he looks forward to working with mathematicians and computer experts to upgrade the graphing method to an algorithm to better serve the public.

2. Successfully found the drawing method of Hamiltonian loop, the judgment law of Hamiltonian diagram, and the approximate mapping method of the traveling salesman problem.

3. Route coloring conjecture.

The route coloring conjecture has been a puzzle that has puzzled the mathematical community for four decades, and was successfully solved by Israeli mathematicians in 2007, but it still has limitations.

You can try it out and let everyone experience the fun of mathematics.

This four-color conjecture has not been proved by Tachibana Tanhe in the strict sense.

There are mathematicians who make use of computers. It has been proven, but some mathematicians still do not recognize this method.

Addendum: Computer Proofs of the Four-Color Problem.

The invention of high-speed digital computers prompted more mathematicians to study the "four-color problem". Heiko, who had been studying the four-color conjecture since 1936, openly claimed that the four-color conjecture could be proved by finding an inevitable group of reducible figures. His student Trey wrote a computational program that Heiko could not only use to prove that the configuration was reducible, but also by modifying the map into what is mathematically called "dual".

He marked the capitals of each country, and then connected the capitals of the neighboring countries with a railroad that crossed the border, erasing all the lines except the capitals (called the vertex) and the railroad circles (called the letter dust arcs or edges), and the rest was called the dual diagram of the original map. By the late sixties, Heiko had introduced a method similar to moving charges in an electrical network to find the inevitable group of configurations. The "discharge method", which appeared in a rather immature form for the first time in Heiko's research, was a key factor in later research on the inevitable group and a central element in the proof of the four-color theorem.

After the advent of electronic computers, due to the rapid increase in the speed of calculation and the emergence of human-computer dialogue, the process of proving the four-color conjecture was greatly accelerated. Harken of the University of Illinois in the United States set out to improve the "discharge process" in 1970 and later worked with Appel to develop a good program. In June 1976, they spent 1,200 hours and 10 billion judgments on two different electronic computers at the University of Illinois in the United States, and finally completed the proof of the four-color theorem, which caused a sensation in the world.

This is a major event that has attracted many mathematicians and math enthusiasts for more than 100 years, and when the two mathematicians published their research, the local post office stamped all the mail sent out that day with a special postmark of "four colors are enough" to celebrate the solution of the problem.

Let the plot g=(v,e), where v=, and g is a simple undirected planar view. Here g has n vertices and n>3. From the inference of Euler's theorem, it can be seen that there is at least one vertex in the plane graph g, whose degree does not exceed 5.

Let's assume that this vertex is v0. We use mathematical induction, and when n = 4,5, the theorem is obviously true. When deg(v0)=3,4, the theorem is clearly true.

When deg(v0)=5, as shown in the figure, it is not difficult to know that v1 and v3 are not likely to be adjacent to v2 and v5 at the same time, otherwise the planarity of the graph will be destroyed. Let's assume that v1 and v3 are not adjacent. We shrink the three points of v1, v0, and v3 along the edges (v1, v0), (v0, v3).

Another figure g' is obtained, and since this contraction is a continuous change, it does not change the planarity of the graph, i.e., figure g' is still a planar view (as shown in Figure 2). It is obvious that the number of vertices in the figure g' is less than n, so according to the above induction, it can be seen that g' can be colored in five colors: c1, c2, c3, c4, c5.

Let's assume that v2 is c2, v5 is c5, v4 is c4, and the shrinking point is c1 ......Then we can do this for the g, except for the vertices v0, v1, v3, the rest of the vertices have the same shading scheme as g'. For v1 and v3 we can use c1, so that in the diagram g v1, v3 uses c1, v2 uses c2, v4 uses c4, v5 uses c5, then the remaining c3 is on the vertex v0, so g can be colored with five colors. According to the principle of induction, it can be seen that for all floor plans, the five-color theorem is true.

The four-color theorem is a consequence of the coloring problem of graphs. The essence of coloring a graph is to label vertices in a graph, but certain conditions must be met. Color is just a label.

Although the description of the four-color theorem mentions maps, the four-color theorem is not required for mapping: it only needs to be colored, and it does not need to use a minimum of colors. When actually drawing a map, you don't usually use four colors.

Application of coloring problems, mainly scheduling and distribution problems.

For example, I have several tasks, and each task takes a day. I know that several of these tasks are conflicting and cannot be scheduled to be completed on the same day. Now I want it to be done in four days.

This is the four-color problem: the graph used is the vertex of the task, the conflicting tasks are connected to each other, the date is used as the color, and the graph is colored.

Another example is that I have some employees who I want to divide into four groups. But I knew that several of the employees were at odds with each other and couldn't be placed in the same group. Then this is again a four-color problem:

The diagram used is based on the employees as the apex, and the contradictory employees are connected to each other, and the group is used to color and color the diagram.

The four-color theorem says: If the diagram mentioned above is a plane diagram (determined by an efficient algorithm), then it may be completed in four days and may be divided into four groups.

The four-color theorem is a basic theorem of graph theory, from which many theorems can be derived, and the direct application in life is not common. The reason why the four-color theorem is famous is that this simple problem has not been proved relatively simply, which has aroused great interest among mathematicians, and until now, the computer proof is still the most direct and valid proof of the four-color theorem.

The four-color theorem isn't on a map?

At least four colors can be used to distinguish countries from each other.

1.It is applied to some map processing fields and zoning markers, which is one of the three major mathematical problems of contemporary times.

2.There are applications in the processing of many mathematical problems, such as computer science, finance, and mathematics, and there is a four-color theorem.