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, . . . , Cg such that, after cutting along C1, . . . , Cg, we obtain a sphere with g holes and g Möbius strips, an odd number of which is nonbipartite.