Quadrangular embeddings of complete graphs and the Even Map Color Theorem

Hartsfield and Ringel constructed orientable quadrangular embeddings of the complete graph Kn for n≡5(mod8), and nonorientable ones for n≥9 and n≡1(mod4). These provide minimal quadrangulations of their underlying surfaces. We extend these results to determine, for every complete graph Kn, n≥4, the minimum genus, both orientable and nonorientable, for the surface in which Kn has an embedding with all faces of degree at least 4, and also for the surface in which Kn has an embedding with all faces of even degree. These last embeddings provide sharpness examples for a result of Hutchinson bounding the chromatic number of graphs embedded with all faces of even degree, completing the proof of the Even Map Color Theorem. We also show that if a connected simple graph G has a perfect matching and a cycle then the lexicographic product G[K4] has orientable and nonorientable quadrangular embeddings; this provides new examples of minimal quadrangulations.