Gr\"unbaum Colorings of Toroidal Triangulations
  We prove that if G is a triangulation of the torus and \chi(G) \neq 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors. 
Michael O. Albertson, Hannah Alpert, sarah-marie belcastro, and Ruth   Haas