Abstract
The Four Color Theorem is equivalent with its dual form stating that each 2-edge-connected 3-regular planar graph is 3-edge-colorable. In 1968, Grünbaum conjectured that similar property holds true for any orientable surface, namely that each 3-regular graph with a polyhedral embedding in an orientable surface has a 3-edge-coloring. Note that an embedding of a graph in a surface is called polyhedral if its geometric dual has no multiple edges and loops. We present a negative solution of this conjecture, showing that for each orientable surface of genus at least 5, there exists a 3-regular non 3-edge-colorable graph with a polyhedral embedding in the surface.
Supported by grant VEGA 2/7037/7 and by A. v. Humboldt Fellowship.
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Kochol, M. (2009). 3-Regular Non 3-Edge-Colorable Graphs with Polyhedral Embeddings in Orientable Surfaces. In: Tollis, I.G., Patrignani, M. (eds) Graph Drawing. GD 2008. Lecture Notes in Computer Science, vol 5417. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00219-9_31
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DOI: https://doi.org/10.1007/978-3-642-00219-9_31
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