  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.
