Is a cubic bipartite graph embedded on a torus always 3 face colorable? This is true for a planar graph. We can prove it on the dual graph by considering triangulation of an Eulerian graph and then coloring the 3 vertices of a white colored face with 3 colors in clockwise order and anticlockwise on a black colored face. You can find it on page 4 in the following document :
However I am working on a graph embedded on a torus. Is the result true in this case too? If yes, can you please sketch the outline of the proof?