Consider 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.

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Copyright © 2018 Electronic Journal of Combinatorics. Article first published online: March 2018.

The definitive version is available at: https://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p59

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Full Citation:

Dasbach, Oliver T. and Heather M. Russell. "Equivalence of Edge Bicolored Graphs on Surfaces." Electronic Journal of Combinatorics 25, no. 1 (March 2018): 1-15. https://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p59