1. Hamilton Paths and Circuits

2. A collection of vertices and edges where
Vertex-Edge Graph
A collection of vertices and edges where
each edge connects at least one of the vertices

3. Edges Line segments or curves connecting two vertices
All edges in this graph are shown in red
EDGE

4. Vertices The points where edges meet
All the vertices in this graph are shown in red
VERTEX

5. Hamilton Path A Hamilton Path is a path that
visits each vertex exactly one time.
A Hamilton Path starts at one vertex
and ends at a different vertex.
Hamilton Path

6. Hamilton Path
Start
End

7. Hamilton Circuit A Hamilton Circuit is a path that
visits each vertex exactly one
time.
A Hamilton Circuit starts and ends
at the same vertex.
Hamilton Circuit 7

8. Hamilton Circuit
Start
End

9. How to Describe a Hamilton Circuit or Path
End
Start A B C D E F G H I A
> H
> I
> G
> F
> C
> E
> D
> B
> A

10. A Common Trick Question on a Test
Is this description of a Hamilton Path or Circuit correct? A
> H
> I
> B
> F
> C
> E
> D
> A

11. A Common Trick Question on a Test
The Answer – it’s incorrect
Vertex B is listed twice! A
> H
> I
> B
> F
> C
> E
> D
> A
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.

12. Practice Identifying Hamilton Paths and CircuitsB E F D G A C
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? 12

13. Practice Identifying Hamilton Paths and CircuitsB E F D G A C
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
PATH Only

14. Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? B A K F H G J
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? 14

15. Practice Identifying Hamilton Paths and CircuitsB A K F H G J
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: Path at only A, B, J
PATH Only (must start or end at A)

16. Is there a Hamilton Circuit and or Path in this Vertex-Edge Graph? B C E D F H J G K
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? 16

17. Practice Identifying Hamilton Paths and CircuitsB C E D F H J G K
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