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?
I found a counter example for this. There can be bipartite cubic 3-edge colorable graphs embedded on a torus and yet not 3-face colorable. The dual of the graph given in the beginning of this document is a counter example :
http://www.math.jhu.edu/~jmb/note/torustri.pdf
Editing this question with the answer for the benefit of others.
Thank you!!
Vinuta