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Discrete Mathematics with Graph Theory (3rd Edition) 250

# Discrete Mathematics with Graph Theory (3rd Edition) 250 -...

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248 Solutions to Exercises 5. (a) [BB] Beta index = ~; (b) Beta index = ~ = 1. 6. [BB] 7. The other couples are labeled Al and A 2 , Bl and B 2 , C 1 and C 2 . You shook three hands; your partner shook three hands. 8. (a) Let A be any vertex and consider the five edges which meet at A. At least three of these have the same color. Suppose these edges are AB, AC, AD and that these are all red. If any of the edges BC, BD, CD is red, we would have a red triangle; for example, if BC were red, triangle ABC would be red. On the other hand, if BC, BD and CD are all white, then triangle BCD is white. In any case, we have a monochromatic triangle. (This exercise is a restatement of the mutual friends/mutual strangers problem, Problem 17, discussed in Section 6.3.) A ~ /ID // I .(.:: .................... C B (b) As shown to the right above, the result of (a) is not true for a graph with five vertices. 9. [BB] If there are four or more vertices of one color, say red, then there are at least (~) = 4 red triangles.
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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 color---call 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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