Discrete Mathematics with Graph Theory (3rd Edition) 275

Discrete - Section 10.2 273 14 The reader will appreciate that it is difficult to insert three-dimensional models within the covers o f a book

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Section 10.2 273 14. The reader will appreciate that it is difficult to insert three-dimensional models within the covers of a book! 15. (a) [BB] As suggested, add an extra vertex v to g and join it to all other vertices. Then deg v = n ~ nt l , and deg w ~ n2l + 1 = ~ for all other vertices. By Theorem 10.2.4, this new graph with n + 1 vertices has a Hamiltonian cycle. Deleting v and all the new edges incident with v leads to a Hamiltonian path in our original graph. (b) The first graph satisfies the condition in (a), so it has a Hamiltonian path. One is shown to the left below. 1 2 (c) The second graph has a Hamiltonian path as shown on the right above. (d) No. The graph on the right above is a counterexample since it has a Hamiltonian path while n2l = ~ = 4! and not all vertices have degree at least 5. Another example is the graph 0-0- 0-0-0 . This certainly has a Hamiltonian path, but n2l = 5 2 1 = 2 and the end vertices are of degree one. Also, the converse to Dirac's Theorem is false, as we can see by the graph on the right above. This
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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