13.2 Euler Paths and Euler Circuits 853 A snowplow driver must plow every road in a township. To minimize the time and cost involved, the driver would like to establish a route that would allow the plow to travel over each road exactly one time. In this section, we will study paths and circuits that can be used to determine the best route for the snowplow driver. Euler Paths and Euler Circuits SECTION 13.2 LEARNING GOAL Upon completion of this section, you will be able to: 7 Solve problems involving Euler paths and Euler circuits. Why This Is Important There are many real-life problems that can be represented and solved using graphs, paths, and circuits. In addition to transportation problems like the one described above, graph theory can be used to solve problems involving city planning, computer architecture, interior design, industrial engineering, and many other fields. 40. Representing a Neighborhood The map of the Temple Hills subdivision is shown below. Use a graph to represent the streets in this subdivision. Use letters to label the vertices. * Lake Lake Lake Spring Blvd. Brook Ave. Creek St. Pond St. Lake St. Sea St. River St. Ocean Blvd. Water St. Stream Circle Challenge Problems/Group Activities 41. Class Poll Poll your entire class to determine which students knew each other prior to taking this course. Draw a graph that shows this relationship. Answers will vary. 42. Odd Vertices Attempt to draw a graph that has an odd number of odd vertices. What conclusion can you draw from this exercise? It is impossible to have a graph with an odd number of odd vertices. 43. Degrees Draw four different graphs and then for each graph: a) determine the degree of each vertex. Answers will vary. b) determine the sum of the degrees from all vertices from this same graph. Answers will vary. c) determine the number of edges on this same graph. Answers will vary. d) What conclusion relating the sum of the degrees of vertices to the number of edges can you draw from this exercise? Explain why this conclusion is true. * Recreational Mathematics 44. Facebook Friends Read the Recreational Math box on page 844. Belinda, Dennis, Jeff, Kristie, Lucy, and Mike are all on Facebook. The following pairs of people are friends: Belinda and Dennis, Belinda and Kristie, Belinda and Lucy, Dennis and Kristie, Dennis and Lucy, Dennis and Mike, Jeff and Lucy, Kristie and Lucy, and Kristie and Mike. Represent the friendships with a graph. * 45. Home Floor Plan Use a graph to represent a) the floor plan of your home. Answers will vary. b) the streets in your neighborhood within 1 2 mile from your home in all directions. Answers will vary. Research Activity 46. Continent Graph Choose a continent other than North America or Australia and create a graph showing the common boundaries between countries. *See Instructor Answer Appendix Pi-Lens/Shutterstock
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