Equivalence of edge bicolored graphs on surfaces

TL;DR

This paper investigates the classification of edge bicolored graphs embedded on surfaces, counting equivalence classes under face and vertex color reversals, with implications for knot theory.

Contribution

It extends the understanding of edge bicoloring equivalence classes from planar graphs to graphs on arbitrary surfaces, revealing new computational challenges.

Findings

Counted equivalence classes for graphs on various surfaces

Identified differences from planar case complexities

Linked graph colorings to knot theory applications

Abstract

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.