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An embedding of a 3-regular graph in a surface is called polyhedral if its dual is a simple graph. An old graph-coloring conjecture is that every 3-regular graph with a polyhedral embedding in an orientable surface has a 3-edge-coloring. An affirmative solution of this problem would generalize the dual form of the Four Color Theorem to every orientable surface. In this paper we present a negative solution to the conjecture, showing that for each orientable surface of genus at least 5, there exist infinitely many 3-regular non-3-edge-colorable graphs with a polyhedral embedding in the surface.

Polyhedral embeddings of snarks in orientable surfaces