1. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 1 4 Graph Theory (Networks) The Mathematics of Relationships 4

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 2 Graphs, Puzzles, and Map Coloring 4.1 Understand graph terminology Apply Euler’s theorem to graph tracing Understand when to use graphs as models (continued on next slide)

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 4.1, Slide 3 Graphs, Puzzles, and Map Coloring 4.1 Use Fleury’s theorem to find Euler circuits Utilize graph coloring to simplify a problem

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 4 Graph Terminology

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 5 Examples of Graphs http://oak.cats.ohiou.edu/~ridgely/Volvo_docs/ http://maps.google.com/?ie=UTF8&ll=57.703528,11.966429 &spn=0.017449,0.036993&z=15

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 6 Graph Terminology

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 7 Graph Tracing The Koenigsberg bridge problem Starting at some point, can you visit all parts of the city, crossing each bridge once and only once, and return to the starting point?

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 8 Graph Tracing We can model the problem with a graph model.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 9 Graph Tracing The Koenigsberg problem, phased in graph theory language, is “Can the graph be traced?” To trace a graph means to begin at some vertex and draw the entire graph without lifting the pencil and without going over any edge more than once.

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 10 Graph Tracing *Connected graphs are also called networks.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 11 Graph Tracing Example: (solution on next slide)

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 12 Graph Tracing Example: Odd: B and C Even: A, D, E, and F (Zero edges is considered “even”)

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 13 Euler’s Theorem

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 14 Euler’s Theorem

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 15 Euler’s Theorem Example: Which of the graphs can be traced? (solution on next slide)

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 16 Euler’s Theorem Solution: Not trace-able Trace-able

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 17 Euler’s Theorem

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 18 Euler’s Theorem Example: Solution: Path ACEB has length 3. Path ACEBDA is an Euler path of length 5. It is also an Euler circuit.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 19 Fleury’s Algorithm We use Fleury’s algorithm to find Euler circuits. (example on next slide)

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 20 Fleury’s Algorithm Example: (solution on next 3 slides)

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 21 Fleury’s Algorithm Solution: (answers may vary) (continued on next slide) erase

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 22 Fleury’s Algorithm Solution: (answers may vary) (continued on next slide) erase

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 23 Fleury’s Algorithm Solution: (answers may vary) erase

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 24 Eulerizing a Graph We can add edges to convert a non- Eulerian graph to an Eulerian graph. This technique is called Eulerizing a graph. (example on next slide)

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 25 Eulerizing a Graph Example: (solution on next slide)

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 26 Eulerizing a Graph Solution: (answers may vary) Eulerize

27. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 27 Map Coloring

28. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 28 Map Coloring Example: (continued on next slide)

29. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 29 Map Coloring Solution: (continued on next slide) We can rephrase the map-coloring question now as follows: Using four or fewer colors, can we color the vertices so that no two vertices of the same edge receive the same color?

30. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 30 Map Coloring Solution: There is no particular method for solving the problem. A trial-and-error is shown. Other solutions are possible.

31. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 31 Map Coloring

32. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 32 Map Coloring Example: Each member of a city council usually serves on several committees in city government. Assume council members serve on the following committees: police, parks, sanitation, finance, development, streets, fire department, and public relations. Use the table below to determine a conflict-free schedule for the meetings. We do not duplicate information - that is, because police conflicts with fire department, we do not also list that fire department conflicts with police. (solution on next slide)

33. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 33 Map Coloring Solution: Model the information with a graph. Join conflicting committees with an edge. (continued on next slide)

34. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 4.1, Slide 34 Map Coloring Solution: Using trial and error and four colors or less, color committees that can meet at the same time with a common color. That is, similar-colored vertices should not share a common edge (same problem as the four-color problem).