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 - with
faces on a surface with Euler characteristic
Suppose also that the surface can be coloured with at most
colours where
(1)
Then at least one face must have a boundary with less thanedges.
Proof:
(2)
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 is
then 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
If
is the number of faces with
edges, then
and
so
Hence at least one face must have a boundary consisting of less than six edges.