Variations on Ringelʹs earth–moon problem

We use current graphs to find decompositions of complete graphs into subgraphs with certain embeddability properties. These decompositions provide solutions to various extensions of Ringelʹs earth–moon problem to other surfaces and to more than two surfaces. In particular, we find a decomposition of K6n+1 into n toroidal graphs, and from this get a decomposition of K6n into n projective-plane graphs. We find a decomposition of K19 into three Klein bottle graphs and for n>3, we also conjecture that there exist decompositions of K6n+1 into n Klein bottle graphs. Finally, we find the 3-chromatic number for infinitely many different orientable surfaces.