Another Proof of the Map Color Theorem for Nonorientable Surfaces

The simplest known proof of the Map Color Theorem for nonorientable surfaces (obtained by Youngs, Ringel et al. and given in Ringelʹs 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 Youngsʹ and Ringelʹs 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 ladder-like 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).