- #1
Cuttlas
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Hi
What is the face chromatic number for the Graph Below?
Thank You
What is the face chromatic number for the Graph Below?
Thank You
Face Chromatic Number of A Graph
- Thread starter Cuttlas
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In summary, the conversation discusses the face chromatic number for a given graph. The participants debate whether the number is 4 or 2, and provide explanations and diagrams to support their arguments. Ultimately, it is determined that the face chromatic number is 2, as evidenced by the diagram provided by one of the participants.
- #2
HallsofIvy
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Four.
- #3
Cuttlas
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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?
- #4
willem2
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Cuttlas said:
It can't be the chromatic number of the vertices either, because that's equal to 3.
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.
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- #5
Cuttlas
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willem2 said:It can't be the chromatic number of the vertices either, because that's equal to 3.
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?
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Cuttlas said:
I think willem2 was agreeing with you that the face chromatic number is not 4.
- #7
Cuttlas
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haruspex said:
:) Yes, it seems your right. Then Does it mean I'm right? The Face chromatic number is 2 or 3?
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Your own diagram seems to prove it's 2. Why are you unsure?Cuttlas said:
1. What is the face chromatic number of a graph?
The face chromatic number of a graph is the minimum number of colors needed to color the faces of the graph such that no two adjacent faces have the same color.
2. How is the face chromatic number of a graph calculated?
To calculate the face chromatic number of a graph, you must first determine the maximum degree of any face in the graph. Then, you can use the formula (maximum degree + 1) to find the minimum number of colors needed to color the faces of the graph.
3. What is the relationship between the face chromatic number and the vertex chromatic number of a graph?
The face chromatic number is always greater than or equal to the vertex chromatic number of a graph. This means that the minimum number of colors needed to color the faces of a graph is at least as much as the minimum number of colors needed to color the vertices of the same graph.
4. Can the face chromatic number of a graph be equal to the vertex chromatic number?
Yes, it is possible for the face chromatic number and the vertex chromatic number of a graph to be equal. This can happen in certain types of planar graphs, where the maximum degree of a face is equal to the maximum degree of a vertex.
5. Why is the face chromatic number important?
The face chromatic number is important because it helps us understand the structure and properties of a graph. It can also be used in various applications, such as scheduling and map coloring problems, where minimizing the number of colors used is crucial.
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