Graph Theory (Networks)

1. 4 Graph Theory (Networks) The Mathematics of Relationships

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. Graphs, Puzzles, and Map Coloring 4.1 • Use Fleury’s theorem to find Euler circuits • Utilize graph coloring to simplify a problem

4. Graph Terminology

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. Graph Terminology

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. Graph Tracing • We can model the problem with a graph model.

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. Graph Tracing *Connected graphs are also called networks.

11. Graph Tracing • Example: (solution on next slide)

12. Graph Tracing • Example: Odd: B and C Even: A, D, E, and F (Zero edges is considered “even”)

13. Euler’s Theorem

14. Euler’s Theorem

15. Euler’s Theorem • Example: Which of the graphs can be traced? (solution on next slide)

16. Euler’s Theorem • Solution: Trace-able Not trace-able Trace-able Trace-able

17. Euler’s Theorem

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. Fleury’s Algorithm • We use Fleury’s algorithm to find Euler circuits. (example on next slide)

20. Fleury’s Algorithm • Example: (solution on next 3 slides)

21. Fleury’s Algorithm • Solution: (answers may vary) erase (continued on next slide)

22. Fleury’s Algorithm • Solution: (answers may vary) erase (continued on next slide)

23. Fleury’s Algorithm • Solution: (answers may vary) erase

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. Eulerizing a Graph • Example: (solution on next slide)

26. Eulerizing a Graph • Solution: (answers may vary) Eulerize

27. Map Coloring

28. Map Coloring • Example: (continued on next slide)

29. Map Coloring • Solution: 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? (continued on next 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. Map Coloring

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. Map Coloring • Solution: Model the information with a graph. Join conflicting committees with an edge. (continued on next 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).