Coloring locally bipartite graphs on surfaces

It is proved that there is a function f: ℕ → ℕ such that the following holds. Let G be a graph embedded in a surface of Euler genus g with all faces of even size and with edge-width ≥ f(g). Then (i) If every contractible 4-cycle of G is facial and there is a face of size > 4, then G is 3-colorable. (ii) If G is a quadrangulation, then G is not 3-colorable if and only if there exist disjoint surface separating cycles C₁, ⋯, C_g such that, after cutting along C₁, ⋯, C_g, we obtain a sphere with g holes and g Möbius strips, an odd number of which is nonbipartite. If embeddings of graphs are represented combinatorially by rotation systems and signatures [5], then the condition in (ii) is satisfied if and only if the geometric dual of G has an odd number of edges with negative signature.