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Unformatted text preview: 6 }, {N 5 }. R B R (c) The number oftests required would be @ = 3. N4 N5 N6 5. (a) [BB] "" need not be reflexive. Consider (b) "" is symmetric. If A "" B, then there could be a short between A and B, hence a short between B and A, so B "" A. (c) "" is not antisymmetric. Consider (d) "" is not transitive. Consider 6. (a) [BB] N2 ~ N3 Ng N4 Ns N5 N7 N6 N2 7. (a) @ Ng N4 Ns N5 N7 N6 A , where A "" B and B "" A. B Here, A "" B, B "" C, but A f C. (b) Since g contains triangles, xW) 2: 3. On the other hand, the sets {N l , N 5 , N7 }, {N 2 , N4, Ns} and {N 3 , N 6 , N g } define a 3coloring and hence a partition of the nets. SoxW) = 3. (b) Now we have xW) 2: 4, since Nlo N 2, N 4 , N g determine K4. A 4coloring, and hence a partition of the nets, is {N l , N 5 }, {N 2 , N7}, {N4 ,N 6 ,Ns}, {N g }, so xW) = 4....
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 Summer '10
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 Graph Theory

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