Combinatorics and Graph Theory


A Written Proof of the Four-Colors Map Problem

Authors: Zhang Tianshu

A contact border of two adjacent figures can only be two adjacent borderlines. Let us consider the plane of any uncolored planar map as which consists of two kinds’ parallel straight linear segments according to a strip of a kind alternating a strip of another, and every straight linear segment of each kind consists of two kinds of colored points according to a colored point of a kind alternating a colored point of another, either kind of colored points at a straight linear segment is not alike to either kind of colored points at either adjacent straight linear segment of the straight linear segment. Anyhow the plane has altogether four kinds of colored points. At the outset, we need transform and classify figures at an uncolored planar map. First merge orderly each figure which adjoins at most three figures and an adjacent figure which adjoins at least four figures into a figure. Secondly merge each tract of figures which adjoin at most three figures and an adjacent figure into a figure. After that, transform every borderline closed curve of figures which compose directly the merging figure into the frame of a rectangle which has only longitudinal and transversal sides, according to the sequence from outside merging figure to inside merging figure. Finally color each figure with a color according to either a color of some particular points of a rectangular borderlines closed curve of the figure, or a color unlike colors of its adjacent figures.

Comments: 21 Pages.

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Submission history

[v1] 2014-01-17 19:44:31

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