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3-Regular Non 3-Edge-Colorable Graphs with Polyhedral Embeddings in Orientable Surfaces

  • Martin Kochol
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

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.

Keywords

Planar Graph Dual Form Orientable Surface Multiple Edge Parallel Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Martin Kochol
    • 1
  1. 1.MÚ SAV, Štefánikova 49, 814 73 Bratislava 1, Slovakia and FPV ŽUSlovakia

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