Discrete Mathematics with Graph Theory (3rd Edition) 288

Discrete Mathematics with Graph Theory (3rd Edition) 288 -...

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286 Chapter 10 Review 1. The graphical representation of the new problem is shown at the right. Since the degree of every vertex is now even, Theorem 10.1.4 tells us that the connected pseudograph is Eulerian. So the answer is yes. 2. Recall the graphical representation of the Konigsberg Bridge Problem shown at the right. There are four vertices of odd degree. Building a single new bridge must still leave two vertices of odd degree and the resulting graph will still not be Eulerian. Our mayoralty candidate is not telling the truth! Solutions to Review Exercises C A~------1)B D 3. Since ~h has an Eulerian trail, it has exactly two vertices of odd degree, by Theorem 10.1.5. Let v be one of these. Similarly, let w be one of the two vertices of odd degree in ~h. When v and w are joined, the new graph will be connected, and v and w will now be vertices of even degree. Hence the new graph will have exactly two vertices of odd degree and so will possess an Eulerian trail. A
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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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