Properties of the dual planar triangulations

TL;DR

This paper investigates properties of planar triangulations and their duals, examining numeric and matrix characteristics to understand their interrelations and implications for the Four Color Problem.

Contribution

It introduces conjugated planar triangulations and proves new theorems about their properties, especially regarding their numeric and matrix characteristics related to the Four Color Problem.

Findings

The cyclomatic number of dual triangulations depends on the original's number of vertices.

Dual matrixes must meet similar requirements but differ in at least one characteristic.

Restrictions for Four Color Problem are derived from matrix property differences.

Abstract

This article is devoted to the properties of the planar triangulations. The conjugated planar triangulation will be introduced and on the base of the properties, which were achieved by the other authors there will be proved some theorems, which will show the properties of the dual triangulations. Also the numeric properties of the dual planar triangulations will be examined for the sake of understanding the interdependences of the cyclimatic numbers of different graphs between themselves. We'll see how the cyclomatic number of the planar conjugated triangulation depends on the cyclomatic number of the planar triangulation and how its increment depends on the number of the vertexes. These characteristics will be further very important for examining of Four Color Problem. The properties of the dual matrixes will also be examined. We will see that both matrixes on the one hand must meet the equal requirements, but on the other hand we will see that one characteristic cannot be fulfilled. This fact will further form the restrictions for the solution of Four Color Problem.