One, an overview of the
In 1852, the British mathematician Francis. Old Delhi was first put forward four color conjecture: "any a map only 4 kinds of color can make countries with common borders with different color",
In 1976, mathematician Kenneth. Appell and Wolfgang. Hawking USES the method of "discharge" program, with the help of a computer running 1200 hours, for the first time the four color theorem,
But such a simple problem description, serious acosmia with proof of such a complex process, there have been hoping to find more simple, without proof of process with the help of computer,
This paper attempts to a direct, without checking the four-color theorem proving, with the help of computer
Second, key concept
1) normal map
Normal map is to point to: no one country surrounded by other countries, or no more than three countries meeting in 1 point for normal map (also known as three-dimensional figure in wikipedia),
The same number as shown in the above, the state and the location on the map is also the same situation, regular coloring number must be no less than informal map, this paper mentioned the default refers to the normal map,
2) inevitable configuration
Mine in 1878, mathematician Alfred. Kemp (Alfred kemper) released a four-color theorem proving, kemp expounds two important concepts in the proof, has been regarded as the key to four color theorem, the first concept is "inevitable configuration," in any normal map at least one neighbor has 2 and 4 or 5, there is no every country has & gt;=6 neighboring normal map, that is, two, three, four or five neighbours "configuration" is inevitable, each map must contain the four inevitable configuration of one or more,
Prove process refers to the euler's formula (v: vertex - e: + r:=2), it is concluded that there are at least in arbitrary planar map a country, its neighbours number & lt;=5:
Euler's formula (1) : v - e + r=2;
(2) the plane map: 3 * v<=2 * e;
(3) the euler's formula in formula (2) : 3 * r - e>=6;
(4) assumed that each surface maps with an average of article x, are: x * * e r=2;
(5) to (4) into (3) : 6 - x * * r r> Thus=12, the average number of edges: x<6;
Arbitrary planar maps, therefore, must be at least one country, its neighbours number & lt;=5;
3) configuration can be about
Kemp another concept is "about", namely the inevitable configuration (two, three, four, five neighboring configuration) center of national do any subtract, restore operation, will not increase the number of color map,
Continued to subtract the inevitable configuration state, in the center of the map in the rest of the & gt;=6 neighboring configuration can reduce the number of neighbouring countries and the rest of the country will gradually decline, will eventually have a number of countries is limited, and only contain inevitable topography map, if down to the final map is about four color enough, so do reverse recovery operations to subtract, maps, return to original state map will be four color enough,
Two, three neighboring configuration some reduction, restore regional center of the country will not affect other regional map, the shaded, was about minus four neighboring configuration in recovery, may affect other areas of the map coloring, but kemp transformation can be used to ensure does not increase the number of color map,
Kemp described in the paper five neighboring configuration area also must be reducible, as shown in the figure below, if the configuration area after subtract five neighbours have filled 4 kinds of color, when there is no A - C, or does not exist A - D kemp chain, can be A color change to C, and D color to restore five neighboring configuration, when there is A - C at the same time, A - D two chain, may be brought back five neighboring configuration as follows (five neighboring configuration area 6 countries with 4 color, and not to increase the number of color map to other area),
4) when wood counterexample and five-color theorem
In 1890, Percy. John wood (Heawood) pointed out the defects in "kemp chain" method, and constructs the unusable kemp chain transform the local five neighboring configuration of the 4 color map instance, called Mr Wood counter example,
At least 16 countries of wood structure counterexample diagram, target five neighboring configuration around 5 neighboring countries, there is A - C and A - D two "cross" kemp chain, to the local topography map around five neighbors use coloring kemp chain transformation, 4 kinds of color in the configuration around five neighbours that fit cycle, not to 3 kinds of color, to prove that the five neighboring configuration area (A total of six countries) can be A 4 color,
For this defect, kemp fails to repair strategy is given, so cherish, did not find four color conjecture, but relatively weak five-color theorem is plenteous thesis,
Conjecture, and wood and in arbitrary orientable surface, most only
color dyeing can random maps, k is the surface of the losses, when k=0, the conjecture becomes a four color conjecture," enclave "is in the local figure (map in one of two adjacent areas belong to the same country, must be the same color), k 1 or higher, in 1968, Steven gerrard. And J.W.T. young, finally proved the suspect in the case of k 1 or higher, Until 1976, when a mathematician Kenneth. Appell and Wolfgang. Hawking USES the method of "discharge" program, 1200 hours of verification through the computer, for the first time to get a complete proof, four color conjecture is finally become four color theorem,
This is the first major prove the theorem with the help of computer, was not widely accepted at the beginning, because the public think that cannot be directly validate, although with the popularity of computer, the computer aided prove more acceptable, but some of us want to be able to find a simple, no proof process with the help of computer,
Three, prove process
Below for wood loss was in case of 1 map constructed seven motioned for two adjacent countries,
Draw lessons from the method, we attempt to analyze five neighboring configuration of a minimum number of coloring,
If A five neighboring configuration area in the map manage to make do 4 color (include center, and A total of six countries around), then the five neighboring configuration around five countries only the 3 colors (hereinafter referred to as the "five adjacent three color"), A total of five kinds of equivalent form (below 3 color: A, B, C),
When configuration around five countries in specified order with 3 color (A, B, C), can be easily constructed must use 5 kinds of color map:
But adjust the configuration around five national coloring order (for A, B, C three kinds of color), we found that the map of adjacency relations between countries will cut, after the adjustment can color map with 4 kinds of color, the
This "Oriental not bright western bright, dark south, north" phenomenon, in the configuration around five countries also exists in specified order with 4 color (regardless of the color on the configuration of the center for countries, hereinafter referred to as the "five neighboring four color") :
Possibility: local five neighboring configuration map more than four countries outside, and the configuration specified "5 adjacent 3 color", may not be able to construct with 5 maps of two neighbouring countries, if you can prove this point, also can prove that five neighboring configuration can be some, did not have this point the thorough analysis, this article
According to the five color theorem for arbitrary planar map of local five neighboring configuration, without considering the configuration of the center for national coloring, the configuration around five neighbours, must have the 4 color, because the configuration around a total of 5 "5 adjacent 4 color" equivalent combination, the five kinds of equivalence in the map, there are three kinds of map at least four color enough color scheme:
Inferred: we know the four neighboring configuration must be reducible, four countries around four neighboring configuration also must have the 3 color center (regardless of the configuration of country), because the configuration around a total of 2 "4 adjacent 3 color" equivalent combination, then the two equivalent portfolio, at least one kind of map is a four color enough,
And configuration around the "five neighboring four color" equivalent of 5 kinds of maps, there is a cross spent to repair relationship each other, according to "four adjacent three color" two equivalent combination shall be 1 kind of map four-color enough concluded that "five neighboring four color" five kinds of equivalence in the map, there must be no less than 5/2, namely & gt;=3 kind of map is a four color enough,
If you want to prove that contains five neighbouring topography map manage to make do 4 color, "5 adjacent 3 color" around the configuration of 5 kinds of equivalence in the map, there must be at least 1 kind of map is four color enough,
Namely: if any map local five neighboring configuration "5 adjacent 3 color" five kinds of equivalent portfolio, there are at least 1 kind of map is four color enough, five neighboring configuration must be reducible,
Here to solve any 5 neighboring configuration "5 adjacent 3 color" equivalent of 5 kinds of maps, the least number of coloring:
For four configuration must be reducible to our neighbours, we from the various evolution of four neighboring configuration, analysis of the configuration around all sorts of color combination map minimum number of coloring:
, as shown in the above four countries around when the configuration of the 2 color (hereinafter referred to as "four adjacent two color"), the original map map doesn't seem to have the minimum quantity of staining with evolution;
As shown in the above, and when configuration four countries around 4 color (without considering national coloring of configuration center, hereinafter referred to as the "four neighboring four color"), the original with the evolution of map is the minimum quantity of coloring the following pattern:
Configuration around "4 adjacent 2 color," the original map with evolution of discontinuity, the minimum quantity of coloring analysis because configuration area there are one or a few countries, these countries homogeneous form are neighboring countries around there is no chain of kemp, and limited cannot reuse of neighbouring countries not coloring;
For configuration around 5 adjacent 3 color "" five neighboring configuration, configuration area outside does not exist in similar circumstances, if a country homogeneous form around without neighboring countries have no kemp chain, then can construct kemp chain, or with a configuration not neighboring countries around the same color,
So configuration around "5 adjacent 3 color," turned into "five neighboring four color", the original map with the evolution of chromatic number is continuous, at least have the following inference:
Configuration around "5 adjacent 3 color" equivalent of 5 kinds of maps, and "five neighboring four color" five kinds of equivalent map is as follows: the evolution of the relationship between
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