Discrete Mathematics with Graph Theory (3rd Edition) 368

Discrete Mathematics with Graph Theory (3rd Edition) 368 -...

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366 2. (a) [BB] We show the graph superimposed over the given map. Since this graph contains triangles, at least three colors are necessary. A 3-coloring is shown, so the chromatic number is 3. Solutions to Exercises (b) [BB] False! The Four-Color Theorem says that the chromatic number of a planar graph is at most 4. The planar graph in (a) has chromatic number 3. 3. [BB] In neither case, will four colors necessarily suffice as the pictures to the right demonstrate. (a)~ 1 2 5 4 3 (b)~
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Unformatted text preview: 5 5 3 4 2 4. (a) A 4-coloring is shown; hence, X(Q) ~ 4. Since JC 4 is a subgraph, X(Q) = 4. (b) [BB] A 5-coloring is shown; hence, X(Q) ~ 5. Since JC 5 is a subgraph, X(Q) = 5. (c) A 3-coloring is shown; hence, X(Q) ~ 3. Since JC 3 is a subgraph, X(Q) = 3. (d) A 3-coloring is shown; hence, X(Q) ~ 3. Since Q contains JC3 as a subgraph, X(Q) = 3. (e) [BB] A 4-coloring is shown; hence, X(Q) ~ 4. Since Q contains JC4 as a subgraph, X(Q) = 4. 1 2 1 m 3 4 2 '~ la 2 1 2 1~ at2tV' 2 1 2...
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