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Polyhedral embeddings of snarks in orientable surfaces

Author: Martin Kochol
Journal: Proc. Amer. Math. Soc. 137 (2009), 1613-1619
MSC (2000): Primary 05C15, 05C10
Published electronically: November 14, 2008
MathSciNet review: 2470819
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Abstract: 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.

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Additional Information

Martin Kochol
Affiliation: Suché mýto 19, 811 03 Bratislava 1, Slovakia

Keywords: Edge-coloring, orientable surface, polyhedral embedding, snark, nowhere-zero flow, dual of a graph
Received by editor(s): December 20, 2007
Received by editor(s) in revised form: July 11, 2008
Published electronically: November 14, 2008
Additional Notes: This work was supported by an Alexander von Humboldt Fellowship and by grant VEGA 2/7037/7
Communicated by: Jim Haglund
Article copyright: © Copyright 2008 American Mathematical Society

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