Another proof of the map color theorem for nonorientable surfaces

• Authors: Vladimir P. Korzhik
• Venue: Journal of Combinatorial Theory Series B
• Year: 2002

The simplest known proof of the Map Color Theorem for nonorientable surfaces (obtained by Youngs, Ringel et al. and given in Ringel's book "Map Color Theorem") uses index one and three current graphs, and index two and three inductive constructions. We give another proof, still using current graphs, but simpler than Youngs' and Ringel's in several ways. Our proof uses only index one current graphs and no inductive constructions. For every h ≥ 8, h ≠ 9, 14, we construct an index one current graph Γ(h) that yields a minimal nonorientable embedding ψ(h) of Kh. The current graphs Γ(h) have the property that Γ(n) and Γ(n+1) are not too different from each other and share common ladderlike fragments. As a result, the embeddings ψ(n) and ψ(n + 1) have a large number of common faces: it is shown that, as n approaches infinity, for n ≠ 3, 9 mod 12 (resp. n ≡ 3, 9 mod 12) no less than 5/8 (resp. 5/16) of all faces of ψ(n) appear in ψ(n + 1).