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Discrete Mathematics with Graph Theory (3rd Edition) 381

Discrete Mathematics with Graph Theory (3rd Edition) 381 -...

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Chapter 13 15. We must find the chromatic number of the graph shown. Note that 2,3,6,11 determine a subgraph isomorphic to 1C 4 , so we need at least four colors. Let us select R, W, B, G for these vertices, respectively. We try to color the remaining vertices with four colors. Note that 9 is joined to 2, 6 and 11, so it must be W. Since 4 and 7 are not joined to 3 or 9, we may color this W as well. Since 1 and 12 are not joined to 2 or to each other, we may color them R. Since 5 and 8 are not joined to 6 nor to each other, we color them B. Finally, 10 can be colored G. Wg 379 R 12 Since X(g) = 4, the prisoners can be divided as follows: {I, 2, 12}, {3, 4,7, 9}, {5, 6, 8}, {1O, 11}. 16. The line-of-sight graph 9 will have chromatic number 3. To see why, note that N 3 , N7 and Ns must be assigned different colors, say R, B, G, respectively. Then N2 and N5 can be colored G, and N1 and N4 can be colored B. Finally, N6 can be colored R. The partition of nets is {Nl. N4, N7}, {N 2 ,N 5 ,N s}. {N 3 ,N 6}.
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