A simple proof of the Map Color Theorem for nonorientable surfaces

Abstract

We give a more simple proof of the Map Color Theorem for nonorientable surfaces that uses only four constructions of current graphs instead of 12 constructions used in the previous proof. For every i=0,1,2,3, using an index one current graph with cyclic current group, we construct a nonorientable triangular embedding of K12s+3i+1 that can be easily modified into a nonorientable triangular embedding of K12s+3i+3 and a minimal nonorientable embedding of K12s+3i+5. As a result, for every n≥10, n∉{11,14,20}, we construct a minimal nonorientable embedding of Kn. During the modifications, all but 2(4s+i) faces of the nonorientable triangular embedding of K12s+3i+1 become faces of the nonorientable triangular embedding of K12s+3i+3, and all but 2(4s+i)+1 faces of the nonorientable triangular embedding of K12s+3i+3 become faces of the minimal nonorientable embedding of K12s+3i+5.