# 用吴方法解决染色问题Solve the Dyeing Problem by Wu’s Method

Abstract: Dyeing problem is a famous problem in graph theory; the dyeing problem has been solved by Groebner bases method. This text applies the technique of Wu’s method to solve the dyeing problem. The Wu’s method is also called the feature list method; it is a method to deal with polynomial algebra problem proposed by Wu Wenjun in the 1970s. The difference between Wu’s method with Groebner bases method is that Wu’s method completely uses the viewpoint of zero set to deal with the problem, so it is more effective than Groebner bases method in solving dyeing problems; this paper only solves the 3-color problem in the dyeing problem.

1. 引言

${x}_{i}^{3}-1=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n$(1.1)

${x}_{i}^{2}+{x}_{i}{x}_{j}+{x}_{j}^{2}=0$(1.2)

2. 预备知识

2.1. 升列的定义

1) $cl\left(F\right)

2) $cl\left(F\right)=cl\left(G\right)=p>0$，但 ${\mathrm{deg}}_{{x}_{p}}\left(F\right)<{\mathrm{deg}}_{{x}_{p}}\left(G\right)$

$A:{A}_{1},{A}_{2},\cdots ,{A}_{r}$

1) $r=1,{A}_{1}\ne 0$

2) $r>1,0，且对任意 $j>i,{A}_{j}red/{A}_{i}$

2.2. 矛盾列的定义

2.3. 基列的定义

$B:{B}_{1},{B}_{2},\cdots ,{B}_{s}$

1) 存在 $j\le \mathrm{min}\left(r,s\right)$，使得 ${A}_{1}~{B}_{1},\cdots ,{A}_{j-1}~{B}_{j-1}$，但 ${A}_{j}\succ {B}_{j}$

2) $s>r$${A}_{1}~{B}_{1},\cdots ,{A}_{r}~{B}_{r}$

2.4. 特征列的定义

2.5. 多项式组的零点分解

3. 主要结论

3.1. 吴方法解决染色问题的过程

$\begin{array}{c}{x}_{i}^{2}+{x}_{i}{x}_{j}+{x}_{j}^{2},\left(i,j\right)\in \left\{\left(1,2\right),\left(1,5\right),\left(1,6\right),\left(2,3\right),\left(2,4\right),\left(2,8\right),\left(3,4\right),\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(3,8\right),\left(4,5\right),\left(4,7\right),\left(5,6\right),\left(5,7\right),\left(6,7\right),\left(7,8\right)\right\}\end{array}$

$\begin{array}{l}CS=\left\{{x}_{1}-{x}_{7},{x}_{2}+{x}_{7}+{x}_{8},{x}_{3}-{x}_{7},{x}_{4}-{x}_{8},{x}_{5}+{x}_{7}+{x}_{8},\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{6}-{x}_{8},{x}_{7}^{2}+{x}_{7}{x}_{8}+{x}_{8}^{2},{x}_{8}^{3}-1\right\}\end{array}$

Figure 1. The 3-color figure

$zero\left(PS\right)=zero\left(CS/J\right)+zero\left(PS,{I}_{1}\right)+\cdots +zero\left(PS,{I}_{8}\right)$

3.2. 吴方法和Groebner基方法在解决染色问题时的比较

Groebner基方法 [4][5][6]在计算过程中会使多项式对增加很快，选取多项式对的无序性也会造成结果不同，项序的选择和终止条件的复杂性与否都会影响计算的效率。吴方法 [7][8]在计算多项式方程组时效率较高，速度也很快，整体来说在解决染色问题上吴方法比Groebner基方法更快速和有效。

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https://doi.org/10.1007/978-1-4612-4344-1

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https://doi.org/10.1090/coll/033

[8] Wu, W.-T. (2000) Mathematics Mechanization. Science Press and Kluwer Academic Publish-ers.

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