  The Four color problem is closely related to other branches of mathematics
and practical applications. More than 20 of its reformulations are known, which
connect this problem with problems of algebra, statistical mechanics and
planning. And this is also typical for mathematics: the solution to a problem
studied out of pure curiosity turns out to be useful in representing real
objects and processes that are completely different in nature. Despite the
published machine methods for combinatorial proof of the Four color conjecture,
there is still no clear description of the mechanism for coloring a planar
graph with four colors, its natural essence and its connection with the
phenomenon of graph planarity. It is necessary not only to prove (preferably by
deductive methods) that any planar graph can be colored with four colors, but
also to show how to color it. The paper considers an approach based on the
possibility of reducing a maximally flat graph to a regular flat cubic graph
with its further coloring. Based on the Tate-Volynsky theorem, the vertices of
a maximally flat graph can be colored with four colors, if the edges of its
dual cubic graph can be colored with three colors. Considering the properties
of a colored cubic graph, it can be shown that the addition of colors obeys the
transformation laws of the fourth order Klein group. Using this property, it is
possible to create algorithms for coloring planar graphs.
