Locally planar toroidal graphs are 5-colorable
Michael O. Albertson, Walter R. Stromquist · Proceedings of the American Mathematical Society · 1982
If a graph can be embedded in a torus in such a way that all noncontractible cycles have length at least 8, then its vertices may be 5 5 -colored. The conclusion remains true when some noncontractible cycles have length less than 8, if the exceptions are all homotopic. Essentially this hypothesis means that small neighborhoods of the graph are planar. No similar conclusion holds for 4 4 -colorability.