  Consider the collection of edge bicolorings of a graph that is cellularly
embedded on an orientable surface. In this work, we count the number of
equivalence classes of such colorings under two relations: reversing colors
around a face and reversing colors around a vertex. In the case of the plane,
this is well studied, but for other surfaces, the computation is more subtle.
While this question can be stated purely graph theoretically, it has
interesting applications in knot theory.
