四色定理论证的关键

一百多年来对"四色问题"的研究长期不得其解的关键在于:肯泊(A.Kempe)当年提出的"不可避免构形集"中一个国家(地域)具有五个邻国(邻域)的所谓"可约性"问题得不到解决。"《四色定理》论证"用数学归纳法,而"平面图的点着色方法"未用数学归纳法,两种方式论证"四色问题"都涉及到"一个(待着色)顶点有五个邻接顶点,已着有4种颜色,要将这4种颜色设法变成3种,把腾出来的1种颜色给该顶点着色。"———这就是四色定理论证的关键。再根据换色原理,用巧妙而深层次地换色办法,对这个关键进行更深刻地论述,其换(着)色最多六步就可以完成,进而更充实和完善了前述两文。

A Key to Proof of 4 -Color Theorem

the crucial reason why the 4-color problem has not been solved for hundred years lies in a simplifying problem fails to be resolved. The simplifying problem posed by A.Kempe refers that in the unavoidable structures group, a so-called simplifying problem cannot be solved which happens in the case of that a country or region has five neighboring countries or regions. Mathematical induction was used to proof the 4-color theorem, while "vertex-coloring on a planar graph" (to be issued) does not need to use induction. However both of the methods refer to a main point that a vertex to be colored has five adjacent vertices that are respectively colored 4 colors and the matter is how to free up one color from the four colors to color the vertex. This is also a key to proof the 4-color theorem. So this paper aims to further make the key clear according to the method of interchanging colors to complete the two papers written before. And the steps of interchanging colors are 6 steps at most.