2. Hamilton Paths and Circuits 1 Click to Start

3. 2

4. 3 Start End

5. 4 4

6. 5 Start End

7. 6 Start A H B F G I C D E A> H> I> G> F> C > E > D > B > A End

8. 7 A> H> I> B> F> C > E > D > B > A Is this description of a Hamilton Path or Circuit correct?

9. 8 In a Hamilton PATH each vertex can only be visited once = NO repeating letters In a Hamilton CIRCUIT only the START and END letters may be repeated. The Answer – it’s incorrect A> H> I> B> F> C > E > D > B > A Vertex B is listed twice!

10. 9

11. 10 A Hamilton CIRCUIT does not exist if there is a bridge between two separate circuits Two separate circuits

12. 11 A Bridge connects two separate Paths or Circuits

13. 12 Bridges are One-Way paths BRIDGE - One Way

14. 13 Start End A Hamilton CIRCUIT must have TWO separate Bridges BRIDGE - One Way

15. 14 Start RULE 1: A Hamilton CIRCUIT cannot exist if two separate Circuits are connected by a single Bridge CAN’T RETURN TO START TO START X X

16. 15 RULE 2 A Hamilton CIRCUIT cannot exist when two separate Circuits are connected together at a single Vertex

17. Start End X X X 16

18. Vertex of Degree 1 START RULE 3 A HAMILTON CIRCUIT cannot exist when a Vertex-Edge Graph contains a Vertex of Degree 1 17 One Way A Path Can Exist X X Cannot Continue

19. 18 A C E G K M B C H Vertex of Degree 1 D Existence of a Path with a Vertex of Degree 1

20. 19 A C E D K M B C G H HAMILTON PATH exists Start D G Existence of a Path with a Vertex of Degree 1

21. 20 A C E D G K M B A C E K M B H G Start End HAMILTON PATH exists Existence of a Path with a Vertex of Degree 1

22. 21 A C E D G K M B A C E K M B H Start NO PATH or CIRCUIT Can’t Visit Existence of a Path with a Vertex of Degree 1

23. 22 A C E D G K M B A C E K M B H NO PATH or CIRCUIT Start Can’t Visit Existence of a Path with a Vertex of Degree 1 End

24. RULE 4 NO Hamilton CIRCUIT or PATH can exist when three or more separate Hamilton Paths or Circuits are bridged together at a single Vertex: 23

25. Start X X 24

26. 25 1. Two separate Circuits are connected by one Bridge = NO HAMILTON CIRCUIT (Path Only)

27. 26 Start RULE 1: A Hamilton CIRCUIT cannot exist if two separate Circuits are connected by a single Bridge CAN’T RETURN TO START TO START X X CAN CROSS ONE WAY

28. 27 1. Two Paths or Circuits are connected by only one Bridge 2. Two separate Hamilton Circuits or Paths are connected together at a single Vertex: = NO HAMILTON CIRCUIT (Path Only)

29. Start End X X X 28 CAN’T RETURN TO START

30. 29

31. Vertex of Degree 1 START When A Vertex-Edge Graph contains a Vertex of Degree 1: No HAMILTON CIRCUIT exists (a PATH may exist) X X 30 CAN’T RETURN TO START TO START PATH Circuit?

32. 31 A C E D G K M B A C E K M B H G Start End HAMILTON PATH exists Existence of a Path with a Vertex of Degree 1

33. 32 A C E D G K M B A C E K M B H Start NO PATH or CIRCUIT Can’t Visit Existence of a Path with a Vertex of Degree 1

34. 33 1. Two Paths or Circuits are connected by only one Bridge 2. Two separate Hamilton Circuits or Paths are connected together by a single Vertex: = NO HAMILTON CIRCUIT (Path Only) 3. A Vertex-Edge Graph with a Vertex of Degree 1: = NO CIRCUIT = PATH: Exists Only If the route STARTS or ENDS at the Vertex of Degree 1 4. Three or more separate Hamilton Circuits or Paths are connected together at a single Vertex: = NO HAMILTON CIRCUIT or PATH

35. Start X X 34 Can’t revisit vertices

36. 35 Student Practice Click to Continue

37. 36 A B D C E G F Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker and describe it using vertex letters Does your answer depend on which vertex you start at? Click to Continue

38. 37 Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker and describe it using vertex letters Example: A > B > C > D > E > F > G Does your answer depend on which vertex you start at? YES: No Path or Circuit if start at D or F 37 PATH Only A D E G F B C Click to Continue

39. 38 Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker and describe it using vertex letters Does your answer depend on which vertex you start at? Why? A B D C H G E J F K Click to Continue

40. 39 Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker or describe it using vertex letters one example: A > K > B > C > D > E > F > G > H > J Does your answer depend on which vertex you start at? If it does, explain how it changes, starting at other vertices. YES: No Path or Circuit if start at K, F, G, or H PATH Only (must start or end at A) A B D C H G E J F K Click to Continue

41. 40 Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker or describe it using vertex letters Does your answer depend on which vertex you start at? A B D C H G E J F K Click to Continue

42. 41 Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? If there is, show it with a color marker or describe it using vertex letters Example: A > B > D > G > H > E > J > K > F > C > A Does your answer depend on which vertex you start at? No Both a CIRCUIT and PATH A B D C H G E J F K Click to Continue

43. 42 Click to Exit