Discrete Mathematics with Graph Theory (3rd Edition) 369

Discrete Mathematics with Graph Theory (3rd Edition) 369 -...

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Section 13.2 (f) A 4-coloring is shown and, in fact, X(Q) = 14. Since ACG is a triangle, these vertices must have different colors. Since AC D and AFG are triangles, D and F must have different colors. But now a fourth color is needed for E, which is adjacent to A, D andF. 5. (a) This graph is bipartite, so X(Q) ~ 2. Since the graph has an edge, X(Q) = 2. (b) A 3-coloring is shown. Since 9 contains triangles, X(Q) = 3. (c) A 4-coloring is shown. Since 9 contains K4 as a subgraph, x(9) = 4. (d) A 4-coloring is shown. Since 9 contains K4 as a subgraph, x(9) = 4. (e) A 4-coloring is shown. To see that
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Unformatted text preview: there is no 3-coloring, start with three colors on a triangle and you will see that eventually a fourth color is required. (f) A 5-coloring is shown. Since 9 contains K5 as a sub graph, X(Q) = 5. 3 3 B~Al 2HG C F 2 2 D 3 4 E 1 1 1 * 2 2 2 1 3 Ck::. ..:e--4i~_~2 2 1 4 1 2 3 4 ~ 1 3 2 2 1 5~3 1 4 367 6. (a) [BB] The graph 91 on the left has a 3-coloring, as shown. Since it contains a triangle, X(91) = 3. The graph 92 on the right has a 3-coloring, as shown. By trying to label alternately the vertices of the outer pentagon, we see that two colors will not suffice, so X(92) = 3 too....
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