Unformatted text preview: Otherwise, there are three red and three white vertices, hence, one triangle of each color. 10. By Exercise 8, we know that the graph contains at least one monochromatic triangle. Select such a triangle and name it ABC. We may assume all its edges are colored white. If there were any any other white triangle, we would be done, so assume D.ABC is the unique white triangle in the graph. Let D be any vertex different from A, B, C. Three edges incident with D have the same colorcall themDX, DY, DZ. If one of the edges XY, YZ, ZX were of the same color as DX, we would have a second monochromatic triangle. So we are finished unless {X, Y, Z} = {A, B, C}. Thus XYZ is white and DA, DB, DC are all red. The last paragraph applies to all three vertices D, E, F which are different from A, B, C. Thus DA, DB, DC, EA, EB, EC, FA, FB, FC are all red. Since D.ABC is the only white triangle, one of DE, EF, F D must be red giving us a red, and a second monochromatic, triangle....
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 Summer '10
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 Graph Theory, triangle

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