3 face colorability of a trivalent bipartite graph embedded on a torus

Question

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 :

https://web.archive.org/web/20160519025149/http://w3.marietta.edu:80/~mmm002/Math350Spr09/Lectures/Chapter6.pdf

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?

Answer 1

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