Face Chromatic Number of A Graph

  1. Hi

    What is the face chromatic number for the Graph Below?

    [​IMG]

    Thank You
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,678
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    Four.
     
  4. Why is it 4? Look at image below, the face chromatic number is 2, so it should not be 4. Could u please explain for me how did u get 4?

    [​IMG]
     
  5. It can't be the chromatic number of the vertices either, because that's equal to 3.

    [​IMG]

    The chromatic number of a map with one ore more closed loops as borders is always 2. You get a bipartite graph with all the nodes that are separated from the outside with an odd number of borders on one side, and those separated by an even number of borders on the other side.
     

    Attached Files:

  6. If It can not be 2, then How I painted the Graph's Faces with only two colors? As you can see Faces in painted graph with different colors is not Adjacent with each other. So it means I did the painting correctly. But I think I'm mistaking. Could u please upload the painted graph which u think it is right?
     
  7. haruspex

    haruspex 14,104
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    I think willem2 was agreeing with you that the face chromatic number is not 4.
     
  8. :) Yes, it seems your right. Then Does it mean I'm right? The Face chromatic number is 2 or 3?
     
  9. haruspex

    haruspex 14,104
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    Your own diagram seems to prove it's 2. Why are you unsure?
     
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