[2] **viXra:1710.0278 [pdf]**
*submitted on 2017-10-24 06:22:09*

**Authors:** Suehwan Jeong, Junho Yeo

**Comments:** 7 pages, Abstract available on English/Korean both, Main text Korean only

The Four Color Theorem(4CT) is the theorem stating that no more than four colors are required to color each part of a plane divided into finite parts so that no two adjacent parts have the same color. It was proven in 1976 by Kenneth Appel and Wolfgang Haken, but here we will prove 4CT without Computer Resources. We can display the picture to color as a graph, using nodes that are the points that represent each figure in the picture, and stects which is lines that links two nodes neighboring each other. We proved that every picture to be colored can be expressed as a liner graph, made up only with nodes and stects that have the shape of straight lines. We will name the triangle made of nodes and stects that contains other nodes and stect, ‘Triangular Convex Cell’. Also, we will call the graph that has the Triangular Convex Cell and also has the form of a convex set, a ‘Triangular Convex Cell graph’. The Euler characteristic in plane graph is 1, so if we regard the numver of the node v, the number of the stect e, and the number of the sides made by nodes and stects f, the equation v-e+f=1is established. Using this, we can derive the result that the graph with the highest total connection strength, which is the number of the connected graphs of each node in the graph, is a triangular-convex graph. Now, we prove the Triangular Convex Cell graph prove The Four Color Theorem regardless of how many nodes there are in the Triangular Convex Cell, or it’s arranged shape. We can use this to colorize a huge Triangular Convex Cell graph in which there are ∞ nodes inside the Triangular Convex Cell, and delete the nodes and the stects of the graph as needed to fit the picture to be colored. Thus, by using this method, it can be seen that the Four Color Theorem holds for all the pictures.

**Category:** Topology

[1] **viXra:1710.0118 [pdf]**
*submitted on 2017-10-10 23:07:27*

**Authors:** Dmitri Martila

**Comments:** 4 Pages.

It is amazing to see, how the problems find their solutions. Even such extremely long as the 1200 pages of the ABC-hypothesis proof of the ``Japan Perelman'', which is needed to be consumed by the most brilliant men to come. And like the first PCs were huge but became compact, the large proofs can turn into very compact ones. \copyright

**Category:** Topology