TOWARDS A TOPOLOGICAL PROOF OF THE FOUR COLOR THEOREM V

Here is an outline of the global proof. See (1, 2). • Make G cobordant to G, the completed dual graph. • Now fill in an interior vertex v0 and connect to G. We have now a ball which has G as boundary. • Clean out S(v0) to make it Eulerian. • Take x1 2 V (G). Its unit sphere S(x1) intersects G in a vertex, edge or triangle. Define U1 = {S(x0)} • Clean out S(x1) to make it Eulerian. Now take an other vertex x2 in S(x0) \ S(x1) and define U = (B(x0) \ B(x1)) \ S(x2). • Clean out S(x2) to make it Eulerian. Continue like this without modifying edges in S(x0), nor edges in G. • We have to complete things in such a way that at any time we have only to complete a region.