Equivalence of Edge Bicolored Graphs on Surfaces
Keywords: Embedded graphs, Checkerboard graphs, Knot theory, Region crossing change, Cycle and cocycle spaces of graphs
AbstractConsider the collection of edge bicolorings of a graph that are 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.