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

Letbe the Euler characteristic of a surface. If a regular map is drawn on the surface withfaces, vertices andedges 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)

Letdenote the number of faces, each face bounded byarcs 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 isand 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 thatbecomes

Hence

At least onefor this to be true so at least one face must be bounded by less than 6 arcs.