Discrete Mathematics with Graph Theory (3rd Edition) 374

Discrete Mathematics with Graph Theory (3rd Edition) 374 -...

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372 Solutions to Exercises (b) [BB] Vl and V3, for example. (d) [BB] We know X(g) 2: 4 since g contains K4 (vertices H, V2, Va, V4). On the other hand, a 4-coloring is given by {Vl, V3}, {V2' V5}, {V4}, {H}. Hence, X(g) = 4. The above 4-coloring is also a partition of the nets. 2. (a) 36 nodes, 19 grid segments, 7 nets. (C)v,~v, ~V6 V5 V4 (b) V l and V 3, for example. (d) We know X(g) 2: 4 since g contains K4 (with vertices H, V2, V3, V4 as in l(d». On the other hand, a 4-coloring, and partition of the nets, is {V 6 , V2}, {Vl , V4}, {V5, V3}, {H}; so X(g) = 4 . . 3. [BB] False. An example is the net pattern : : I for which the line-of-sight graph is 0 o. 4. (a) Nl N2 N3 (b) The graph contains a triangle, so X(G) 2: 3. W R W A 3-coloring is shown, so X( G) = 3. The corresponding partition of nets is {N l , N 3}, {N 2 ,N4,N
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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 3-coloring 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 4-coloring, 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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