1. Euler Graphs Section 6.2

2. 6.2 Euler Graphs 2 Circuit? Path? Non- traversable? A D E C B A D E C B A D E C B End at A End at B Start at A Miss an edge Start at A

3. 6.2 Euler Graphs 3 Circuit? Path? Non-traversable? A E C B G O I F K P N H J M D 16 Vertices 28 Edges L

4. 6.2 Euler Graphs 4 Stump the Prof

5. 6.2 Euler Graphs 5 Graph Vertices Edges Type ?

6. 6.2 Euler Graphs 6 Conclusion Therefore the type of graph is not determined by.. So, what is it determined by?

7. 6.2 Euler Graphs 7 Does the graph have a Euler circuit? path? or neither. What is the degree of each vertex? Click “yes” if you see a pattern. 1.Yes 2.No

8. 6.2 Euler Graphs 8 Make a hypothesis based on your work Verify by filling in last column on next slide

9. 6.2 Euler Graphs 9 Non- traversable Circuit Path Graph Vertices Edges Type ? 5 5 5 9 9 9

10. 6.2 Euler Graphs 10 Euler’s Theorem Let N = the number of vertices in a graph If N = the graph has an Euler Circuit (EC) If N = the graph has an Euler Path (EP) (Must start at an odd vertex) If N = (or more), the graph is Non-Traversable (NT)

11. 6.2 Euler Graphs 11 Solution to the Konigsberg Bridge Problem

12. 6.2 Euler Graphs 12 Can you draw this figure without taking your pencil from the paper and without retracing any line? 1. Yes, and I can start at any vertex 2. Yes, but only if I start at certain vertices 3. No, it can’t be done no matter where I start

13. 6.2 Euler Graphs 13 I see how Euler’s Theorem applies to this problem 1. Absolutely 2. Sort of 3. Not a clue

14. 6.2 Euler Graphs 14 Snow Plowing 1. Circuit 2. Path 3. Non-traversable Union City Dover Paterson Morristown Clifton Hackensack Passaic East Orange

15. 6.2 Euler Graphs 15 “Eulerizing” 1. 1 2. 2 3. 3 4. 4 What is the minimum number of roads that can be removed so that this graph will have an Euler Circuit? Union City Dover Paterson Morristown Clifton Hackensack Passaic East Orange

16. 6.2 Euler Graphs 16 Security Guard Animation

17. 6.2 Euler Graphs 17 Can a security guard make an Euler Circuit starting at the parking lot 1. Yes 2. No Employee Parking Lot N C F DEBA M K J I L HG

18. 6.2 Euler Graphs 18 What is the minimum number of doors that can be removed so that this floor plan will have an Euler Circuit? 1. 1 2. 2 3. 3 4. 4 Employee Parking Lot N C F DEBA M K J I L HG

19. 6.2 Euler Graphs 19 Can you cross each of the borders between pairs of neighboring New England states once and only once and return to the state from which you started? 1. Yes 2. No 3. Sometimes

20. 6.2 Euler Graphs 20 Here is Euler’s graph of Konigsberg. Can you start at some vertex and cross every edge twice and only twice? 1. Yes 2. No 3. Can’t tell A C D B

21. 6.2 Euler Graphs 21 A final question Why didn’t Euler worry about the cases of 1, 3, 5, … odd vertices? Draw a graph with one odd vertex.

22. 6.2 Euler Graphs 22 Hypothesis: Counting vertices, edges, degrees applet

23. 6.2 Euler Graphs 23 If you could draw a graph with exactly one odd vertex, the sum of all the degrees would be. Odd Even

24. 6.2 Euler Graphs 24 Theorem: The sum of the degrees of all vertices of a graph is 2 * ( ). Corollary: The sum of the degrees of all vertices of a graph is always.

25. 6.2 Euler Graphs 25 Can a graph have nine edges of which 4 have degree 2, three have degree 3 and two have degree 4? 1.Yes 2.No

26. 6.2 Euler Graphs 26 Can a graph have nine edges of which 4 have degree 3, three have degree 4 and two have degree 2? 1.Yes 2.No

27. 6.2 Euler Graphs 27 A graph has 6 vertices of degree 3 and 5 vertices of degree 4. How many edges does the graph have? 1.11 2.19 3.38

28. 6.2 Euler Graphs 28 End of 6.2

29. 6.2 Euler Graphs 29 N C F DEBA M K J I L HG Drats! I’m Stuck Employee Parking Lot Start

30. 6.2 Euler Graphs 30 What is the security man trying to do?

31. 6.2 Euler Graphs 31 N C F DEBA M K J I L HG Darn! I got all the doors, but… Start Employee Parking Lot

32. 6.2 Euler Graphs 32 In floor plans the vertices are 1.The rooms 2.The doors

33. 6.2 Euler Graphs 33 N C F DEBA M K J I L HG All Clear!! Employee Parking Lot Start

34. 6.2 Euler Graphs 34 Draw a graph with 4 vertices (all odd) and 6 edges 4 vertices (all odd) and 3 edges

35. 6.2 Euler Graphs 35 Draw a graph with 4 vertices (all even) and 5 edges (loops are edges) 5 vertices (3 even) and 8 edges

36. 6.2 Euler Graphs 36 But

37. 6.2 Euler Graphs 37 Meta - Material