Proof That Any Regular Map on a Plane Surface or the Surface of a Sphere Must Have At Least One Region Bounded By Fewer Than Six Arcs
Let
be the Euler characteristic of a surface. If a regular map is drawn on the surface with
faces,
vertices and
edges then
For any surface of Euler characteristic greater than 0 we have
(1)
Since the map is regular all vertices have order at least 3 so
(2)
From (1) and (2)
Let
denote the number of faces, each face bounded by
arcs so that F_3 is the number of faces bounded by 3 arcs, F_4 is the number of faces bounded by 4 arcs and so on.
The total number of faces is
and the total number of edges is half the sum of the product of the number of each faces with the number of edges to each face (half since each edge is an edge to 2 faces):
so that
becomes
Hence
At least one
for this to be true so at least one face must be bounded by less than 6 arcs.
