Condition on Number of Edges For a Face on a Regular Map
Suppose we have a regular map - so that each vertex has the same order - withfaces on a surface with Euler characteristicSuppose also that the surface can be coloured with at mostcolours where
Then at least one face must have a boundary with less thanedges.
Since the map is regular every vertex has the same order (number of edges entering or exiting the vertex) so if the order of each vertex isthen number of edges is
We must have k>=3 for a regular map so 2E >= 3
Rearrange (2) to give
Now use (2) to give
Ifis the number of faces withedges, thenand so
Hence at least one face must have a boundary consisting of less than six edges.