A $4$-color theorem for toroidal graphs
Hudson V. Kronk, Arthur T. White · Proceedings of the American Mathematical Society · 1972
It is well known that any graph imbedded in the torus has chromatic number at most seven, and that seven is attained by the graph ${K_7}$. In this note we show that any toroidal graph containing no triangles has chromatic number at most four, and produce an example attaining this upper bound. The results are then extended for arbitrary girth.