Section 13.2 369 10. (a) [BB] ~ (b)(BB] ~ (c) [BB] The graph isn't planar, by Kuratowski's Theorem, as the results of (a) and (b) each illustrate. (d) [BB] A 3-coloring is shown. Since the graph contains triangles, fewer than three colors will not suffice. The chromatic number is 3~3 three. 2 1 (e) [BB] The converse of the Four-Color Theorem says that a graph with X ~ 4 is planar. This is not true, as this graph illustrates: The chromatic number is 3, but the graph is not planar. 11. (a) [BB] By PAUSE 7 of this section, for any n, X(Kn) = n and for any m, n, X(Km,n) = 2. Thus, X(K 14) = 14 and X(K 5,14) = 2. (b) A cycle with 38 edges is a cycle with an odd number (39) of vertices. We must start with two different colors R and W. If we try to alternate these, the odd vertices R and the even W, say, then we run into trouble because vertex 39 is adjacent to an even and an odd number vertex (38 and 1). Hence, X(91) = 3. On the other hand, 92 is a cycle with an even number of vertices, so X(2)
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