转发张彧典的文章：四色猜想中“染色困局构形”的4染色（二）

雷明85639720

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4- dyeing of "coloring dilemma configuration" in four-color conjecture
四色猜想中“染色困局构形”的4染色（二）
Yudian Zhang  Lichong Zhang（1633409368@qq.com）
张彧典      张利翀
Party School of Yuxian County, Shanxi Province, Chin
中国山西省盂县党校

In order to verify the correctness of Theorem 3 and its corollary, we use the property theorem of four-color quadrilateral to transform the diagonal chain of any known four-color quadrilateral, so that the geometric structure no longer has ten symmetry. (This transformation is called non-tenfold symmetric transformation), and then the H- coloring program is applied to them to see if they can correct 4 colors. There are 62 transformable four-color vertex quadrilateral diagonal chains (the distribution in four configurations of E family is 17+15+15+15). By transforming diagonal chains one by one, 62 non-tenfold symmetric coloring dilemma configurations can be obtained, and the correct 4- coloring can be obtained by using H- coloring program. Then, according to the classification of reverse dyeing times from less (2 times) to more (16 times), 15 different configurations are obtained, which we call {Zhang configuration}, abbreviated as {Zi, i=1, 2, 3, …, 15}. As shown in fig. 7, in each configuration, diagonal chains shown by dashed lines are replaced by thick black lines, So that the geometric structure of the configuration is no longer ten times symmetrical.

Invert counterclockwise       Invert counterclockwise 3      Invert counterclockwise 4
twice   Invert 16 times          times   Invert 15 times           times   Invert 14 times  
clockwise                              clockwise                                clockwise

Invert counterclockwise 5     Invert counterclockwise 6      Invert counterclockwise 7
times Invert 13 times            times Invert 12 times              times  Invert 11 times
clockwise                               clockwise                                clockwise

Invert counterclockwise 8     Invert counterclockwise 9     Invert counterclockwise 10
times  Invert 10 times           times  Invert 9 times             times   Invert 8 times   
clockwise                               clockwise                              clockwise

Invert counterclockwise 11    Invert counterclockwise12     Invert counterclockwise 13
times  Invert 7 times              times  Invert 6 times               times  Invert 5 times
clockwise                                clockwise                                clockwise

Invert counterclockwise 14    Invert counterclockwise 15    Invert counterclockwise 16
times  Invert 4 times              times  Invert 3 times               times  Invert 2 times
clockwise                                clockwise                                clockwise            
Figure 7:15 non-ten-fold symmetrical configurations
图7:15个非十倍对称构形
为了验证定理3及其推论的正确性，我们利用四色四边形的性质定理，对任意已知四色四边形的对角链进行变换，使几何结构不再具有十折对称性(这个变换叫做非十折对称变换)；然后把H-着色程序应用到他们身上，看他们能不能正确4--染色。有62个可变换的四色顶点四边形对角链(在E族四种构型中的分布为17+15+15+15)。通过对角链的逐一变换，可以得到62个非十折对称着色困局构形，并利用H-着色程序得到正确的4-着色。然后根据颠倒染色次数从少(2次)到多(16次)的分类，得到15种不同的构形，我们称之为{Zhang构型}，缩写为{Zi，i=1，2，3，…，15}。如图7所示，在每个构形中，虚线所示的对角链被粗黑线代替，使得该构形的几何结构不再是十折对称的。
In fig. 7, the number of reverse dyeings in two directions is marked under each configuration. It is not difficult to see that the inversion times of Z1-Z8 and Z15-Z8 in counterclockwise and clockwise directions are symmetrical, that is, the 4- coloring of the configuration is the same from two directions.
在图7中，在每个构形下标记了两个方向上的颠倒染色的数量。不难看出，Z1-Z8和Z15-Z8在逆时针和顺时针方向的颠倒次数是对称的，即从两个方向看构形的4-着色是相同的。
Therefore, according to the symmetry of dyeing reversal times in clockwise and counterclockwise directions, 15 patterns can be reduced to 8 patterns (Z1-Z8) and The number of reverse dyeing increased from 2 to 9 times
因此，根据顺时针和逆时针方向染色反转次数的对称性，15个图案可以减少到8个图案(Z1-Z8)，颠倒染色从2次增加到9次。
When the diagonal chain of each four-color vertex quadrilateral is transformed, the above 15 non-tenfold symmetric coloring dilemma configurations are generated. If you choose all the different combination transformations in 62 diagonal chains, you will get thousands of non-10-fold symmetric configurations. According to Theorem 3, they can also be correctly 4 dyed by H dyeing program.
当变换每个四色顶点四边形的对角链时，生成上述15个非十折对称着色困局构形。如果你在62个对角链中选择所有不同的组合变换，你会得到成千上万个非10折对称的构形。根据定理3，它们也可以被H染色程序正确地4染色。
The inference of Theorem 3 realizes the research direction given by Professor Lin Cuiqin of Tsinghua University in 1996: If it can be proved that 4- coloring can be successfully achieved for any maximal plane graph after finite inverse coloring, it will be a great success and a shock at home and abroad
定理3的推论实现了清华大学林翠琴教授1996年给出的研究方向：如果能证明对于任意极大平面图，在有限次颠倒染色之后，都可以成功地实现4-染色，这将是一个巨大的成功，也是国内外的一个震惊。
5. 4-Coloring proof of two kinds of dyeing dilemma configurations
5：两种染色困局构形的4-染色证明         
Dyeing dilemma configuration can be divided into two categories according to whether its geometric structure has ten times symmetry. One is a 10-fold symmetric configuration (E-family four configurations and their extended configurations), and the other is a non-10-fold symmetric configuration. The 4-staining proofs of these two configurations are given below.
染色困局构形根据其几何结构是否具有十折对称性可分为两类。一种是10折对称构形(E族四构形及其扩展构形)，另一种是非10折对称构形。下面给出了这两种构形的4染色证明。
5.1 The 4-coloring proof of the dilemma configuration of non-tenfold symmetric dyeing
5.1非十折对称染色困局构形的4—染色证明
This problem has been proved by Theorem 3 and its corollary above. Here, just to help readers understand the 4- coloring process of these configurations in detail, the 4- coloring proof of Z15 configuration with the most unidirectional reverse coloring times is given, as shown in Figure 8: the dotted line indicates the known or generated maximum chain, the thick black line indicates the reverse coloring chain, and the arrow indicates the evolution order of the configuration.
这个问题已经被定理3及其推论所证明。这里，只是为了帮助读者详细了解这些构形的4-着色过程，给出了颠倒着色次数最多的Z15构形的4-着色证明，如图8所示:虚线表示已知或生成的最大链，粗黑线表示颠倒着色链，箭头表示构形的演化顺序。      
5.2 4-staining proof of dyeing dilemma configuration with ten-fold symmetry
5.2具有十折对称性的染色困局构形的4-染色证明
We use mathematical induction to prove it.
我们用数学归纳法来证明。

Figure 8:4-coloring proof of z15  
图8：4-z15的着色证明   
5.2.1 Inductive base proof: Because the E- group 4 configuration has a periodic cycle in the process of H dyeing, it cannot be proved that they can be 4- dyed by this method. In the four minimum E configurations, Z1 and Z2 contain characteristic chain A-B rings, and Z3 and Z4 contain characteristic chain C-D rings. Therefore, we use this dyeing property to give a special "Zhang dyeing program", called "Z dyeing program" for short (some called "tangent chain" method [7], some called "closed chain" method 4).
5.2.1归纳基础证明:由于E-群4构型在H染色过程中存在周期循环，不能证明它们可以用这种方法进行4染色。在四个最小E构型中，Z1和Z2包含特征链A-B环，Z3和Z4包含特征链C-D环。因此，我们利用这种染色性质给出一种特殊的“张染色程序”，简称“Z染色程序”(有的叫“切线链”法[7]，有的叫“闭链”法[4])。
The Z- coloring program coloring method of E-family 4 configuration can also be called Theorem 4.
The z- coloring program of E-family 4 configuration can also be called Theorem 4.
E族4构形的Z-着色程序也可称为定理4。
Only the 4- coloring proofs of the configurations represented by Z1 and Z4 are given below.
下面仅给出Z1和Z4代表的构形的4-染色证明。
Detailed proof of E1 is shown in Figure 9:
E1的详细证明如图9所示：

Fig. 9: 4- staining proof of four configurations of E1 periodic transformation
图9:E1周期变换四种构形的4染色证明
If the graph 1)[ or 3)] is known, firstly reverse the coloring of C-D chain (thick black line) outside A-B ring to generate disjoint A-C, A-D (or B-C, B-D) maximal chains, and then reverse the solitary color B(D), B(C)[ or A(D),A（C）],to reduce the vertex color number of the Pentagon to 3[ see fig. 1) or 3)].
如果图1[或3])已知，首先反转A-B环外C-D链(粗黑线)的着色，生成不相交的A-C，A-D(或B-C，B-D)极大链，然后反转孤立色B(D)，B(C)[或A(D)，A(C)]，将五边形的顶点色数减少到3[见图1)或3]。
If the graph 2)[ or 4)] is known,firstly reverse the coloring of C-D chain (thick black line) outside A-B ring to generate a new B-C (or A-D) maximum chain, and then reverse the vertex color[ A(D)) or B(C)] to reduce the vertex color number of Pentagon to 3 [see Figure 2) or 4)].
如果图2[或4])已知，首先反转A-B环外C-D链(粗黑线)的着色，生成新的B-C(或A-D)最大链，然后反转顶点颜色[ A(D))或B(C)]，将五边形的顶点颜色数减少到3[见图2)或4]。
As for the detailed proof of E4, it is enough to give only the 4- coloring proof of the initial configuration, instead of giving all the 4- coloring proofs of four continuous transformation configurations like E1. As shown in fig. 10.       
至于E4的详细证明，只给出初始构型的4-染色证明就足够了，而不用像E1那样给出四个连续变换构型的全部4-染色证明。如图10所示。

Fig. 10. 4-coloring process of initial configuration of E4 configuration
图10，E4构形初始构形的4-着色过程
Firstly, the coloring of A-B chain (thick black line) is reversed outside C-D ring [dotted line in Figure (1)] to generate a new A-C maximum chain [dotted line in Figure (2)], and then the solitary point D[ big black point in Figure (2)] under A-C maximum chain is changed to B color in Figure (3)
首先在C-D环外反转A-B链(粗黑线)的着色[图(1)中的虚线]，生成新的A-C最大链[图(2)中的虚线]，然后将A-C最大链下的孤点D[图(2)中的大黑点]改为图(3)中的B色。
The 4-coloring proof of Figures 9 and 10 shows that Z staining program is feasible.
图9和图10的四色证明表明Z染色程序是可行的。
5.2.2 Inductive hypothesis proof:
5.2.2归纳假设证明:
Assuming that the Z-coloring program is feasible in the E- family configuration with K=16+5n, the Z-coloring program is still feasible when K=16+5(n+1).
The geometric structure of 10-fold symmetry It is characterized by five-pointed stars and pentagons alternately arranged from inside to outside. When we study the vertex coloring of pentagons separated by five-pointed stars, we find the following rules:
假设Z-着色程序在K=16+5n的E族构型中是可行的，当K=16+5(n+1)时Z-着色程序仍然是可行的。
Within or outside the smallest 16-point E-family configuration, the increased pentagonal vertex coloring always appears in the order of ACDCD or BACDA  and ABCDB. This is determined by the coloring of pentagonal vertices from inside to outside.
十折对称的几何结构其特征是由内到外交替排列的五角星和五角星。当我们研究被五角星分开的五角星的顶点着色时，我们发现以下规律:
Among the four configurations of the E family, E1 is a class  and E2, E3 and E4 are a class (at least one of the two intersecting chains passes through the symmetric central color point), so it is enough to select only the extended proofs of E1 and E4.
在E族的四种构型中，E1是一类，E2、E3和E4是一类(两条相交链中至少有一条穿过对称中心色点)，所以只选择E1和E4的扩展证明就足够了。
In the three figures of fig. 11, the solid line shows that when K=16+5n(n=0, 1, 2, 3, ...), and the dotted line represents the components embedded inside (or outside) the solid line graph,
在图11的三个图中，实线表示当K=16+5n(n=0，1，2，3，...)，虚线表示嵌入实线图内部(或外部)的组件，
The whole figure still shows ten times symmetry.
整个图形仍然显示出十折的对称性。

Figure 11(1), E1 configuration expands inward
图11(1)，E 1构形（向内扩展）
Proof: 证明:
(1)        When in E1 internal expansion, as shown in Figure 11(1):
Because the embedded component, whether it is 10-fold symmetrical or not, will not destroy the intersection of the two color chains A-C and A-D in E1, and the characteristic ring A-B still exists, so the Z staining program is still feasible.
(1)当在E1内部扩张时，如图11(1)所示: 因为嵌入的成分，不管是不是10折对称，都不会破坏E1的两个色链A-C和A-D的交集，特征环A-B依然存在，所以Z染色程序还是可行的。                  
(2) When E1 expands outward, as shown in Figure 11(2):
When the vertex of the kth Pentagon (thick dotted line) in fig. 11(2) is dyed as BACDA (or ACDCD), the vertex dyeing of the (k+1)th Pentagon must be ABCDB, which is exactly the same as the vertex dyeing of the outermost two pentagons in E1 (solid line) in the figure.The A-C and A-D chains are the elongation of the A-C and A-D chains in E1 in the extended interval, and the characteristic ring A-B (or C-D) still exists.  When the dyed letter is BACDA, the generated characteristic ring is A-B ring; When the dyed letter is (A)(C)(D)(C)(D), the generated characteristic ring is C-D ring. Z staining procedure is still feasible.
(2)当E1向外扩张时，如图11(2)所示:

Figure 11(2), E1 configuration  expands outward                                 
  图11(2)，E1构形 向外扩展
当图11(2)中第kth个五边形(粗虚线)的顶点染色为BACDA(或ACDCD)时，第(k+1)个五边形的顶点染色必须为ABCDB，这与图中E1(实线)中最外面两个五边形的顶点染色完全相同。A-C链和A-D链是E1 A-C链和A-D链在延长区间内的拉长，特征环A-B(或C-D)依然存在。当染色字母为BACDA时，生成的特征环为A-B环；当染色字母为(A)(C)(D)(C)(D)时，生成的特征环为C-D环。z染色程序仍然可行。         
(3) Because the characteristic chain A-C in E4 configuration passes through the color point C  of symmetry center, this configuration can not be expands inward, but can only be expands outward, as shown in Figure 11(3):
(3)由于E4构型中的特征链A-C经过对称中心的色点C，这种构型不能向内扩张，只能向外扩张，如图11(3)所示（图13见下一页）:
Looking at fig. 11(3), that is, E4, the result is just opposite to that of fig. 11(2) .
看图11(3)，即E4，结果与图11(2)正好相反。
When the coloring of the k th pentagon (thick dotted line) is ACDCD (or BACDA), the vertex coloring of the (k+1) th pentagon must be ABCDB. Like the case in Figure 11(2), the characteristic ring C-D (or A-B) still exists, and the Z coloring program is still feasible.

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