Summary
In this article, graphs and some graph theorems are considered.
A graph is called planar if it can be depicted on a plane so that its edges
do not intersect at internal points. Each planar graph divides the plane into
several parts, which are called faces, which are important elements of a
planar graph.
A graph is planar if and only if it can be depicted on a sphere without
intersecting edges at interior points.
If the number of vertices, edges and faces of a flat graph G is denoted
by the letters v, e and f, respectively, and number of connected components
of the graph is k, then v – e + f = 1 + k. It is the Euler’s theorem.
The main corollary from the Euler’s theorem: if G is a planar graph
without loops and multiple edges, v ≥ 3. Then e ≤ 3v – 6. and (if the graph G
is bipartite) e ≤ 2v – 4.
Let G be a planar graph without loops and multiple
ribs. Then the minimum degree among the vertices of the graphs is ≤ 5.
Let the graph K_5 is the graph with 5 vertices, where each vertex is
connected to each other, and K_3,3 is the graph, where are two sets of three
vertices, and each vertex from one set is connected to each vertex of other,
but vertices inside any set are not connected. Then, K_5 and K_3,3 are not
planar graphs.
Let G be a three-connected flat graph. Then the set of boundaries of
its faces is exactly the set of its non-separating induced cycles.
Planar graphs is an important part of the graph theory. It has
applications not only in mathematics, but also in many other sciences. There
are many problem in the world, when there are only two dimensions, and you
need to place something there, which requires one more.
Planar graphs theory is still in developing. One of the main open
problems in this theory is the planarity verification task: to verify whether a
given graph is planar.
Despite the fact that the theory as a whole has been well discovered,
there are connections with other theories, so this theory can be studied
endlessly.