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Discrete Mathematics with Graph Theory (3rd Edition) 378

Discrete Mathematics with Graph Theory (3rd Edition) 378 -...

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376 Solutions to Review Exercises 15. [BB] Deleting vertex 2 (and the three edges incident with 2) leaves lC 5• Alternatively, {I, 4, 6}, {2, 3, 5} are bipartition sets for lC 3,3 after deleting edges 14, 16,46 and 35. 16. [BB] Since the graph is not lC n for any n, nor an odd cycle, Brooks's Theorem says X(Q) :::; Do(Q). Here Do(Q) = 3, so we have X(Q) :::; 3. But this graph contains cycles of odd length, so X(Q) ;:::: 3. We conclude that X(Q) = 3. Chapter 13 Review 1. (a) The graph is drawn first planar and then also with straight edges. k (b) There are eight regions, numbered 1,2, ... ,8 with boundaries agk, abh, bc1!, dem, ef j, ghij, cdi and fkim respectively. (c) E = 13, V
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Unformatted text preview: = 7, R = 8, N = 26, V -E + R = 7 -13 + 8 = 2, N = 26 :::; 26 = 2E and E = 13 :::; 15 = 3V - 6. 2. (a) This is not planar. The graph is homeomorphic to lC 5 ("removing" the two vertices of degree 2). (b) This is not planar. Labeling the graph as shown, the subgraph obtained by removing edges BH, CD and D H is homeomorphic to lC3,3: the bi-partition sets are {B,E,G} and {C,D,F}, with vertices A and H "removed". (c) The graph is planar, as shown. A (d) The graph contains a subgraph homeomorphic to lC3,3 as shown. Deleting the edges shown with dotted lines gives the sub graph. What remains would be lC 3,3 except for the additional vertex at G. G I c ---'-_7"'D E c E H D A B c...
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