1. Hamiltonian Circuits and Paths
Section 4.5

2. Sir William Rowan Hamilton
Irish Mathematician
Knighted at the age of 30
Hamiltonian Mechanics are used to determine orbital trajectories of satellites.
Came up with a game…

3. Icosian Game Vertices were meant to represent major cities in Europe.
Goal is to visit all 20 cities.
Let’s PLAY!!!!!!

5. Street Example
Remember when we had the street network and we wanted to inspect each street exactly once? Well now we need to inspect the fire hydrants (they are located at the intersections. What are the intersections on the graph?
Can we start at the garage and stop at each intersection exactly once and return to the garage?

6. The street network a d b f j h i G

7. Hamiltonian Paths and Circuits
Hamiltonian path: You cross each vertex exactly once and start and end at different places.
Hamiltonian circuit: You cross each vertex exactly once and start and end at the same place.

8. Find the Hamiltonian Circuit

9. Find the Hamiltonian Circuit

10. Find the Hamiltonian Circuit

11. So is there an easy rule????
There is not an easy rule, but there is a vague one.
If a connected graph has n vertices, where n > 2 and each vertex has a degree of at least n/2, then the graph has a Hamiltonian circuit.

12. Does it work? We are guaranteed to have a Hamiltonian Circuit.
Downfall: That doesn’t mean we now know how to find it.

13. Does it work?
We are not guaranteed a Hamilton circuit, but there is one.
Downfall: We don’t know if there is a Hamiltonian circuit.

14. Find the Hamiltonian Circuit
We are not guaranteed that there is a Hamiltonian circuit.
We can’t tell by the theorem that there is not one. We had to use trial and error.

15. State which graphs are guaranteed to have a Hamiltonian circuit

16. Let’s do another online quiz…

17. Tournaments
By using directed edges, we can indicate winners and losers in a tournament.
An arrow from A to B means that A beats B.
We use Hamiltonian circuits with digraphs to determine who wins a tournament.

18. Example
List the winners of each game. A B C

19. Why tournaments are cool!
You can give a ranking for each team if there is exactly one Hamilton path.
The winners are listed in order of the vertices of your Hamiltonian path.

20. Example
Draw a digraph for the following school soccer team round-robin tournament.
Find the Hamiltonian path and determine the ranking.
Game AB AC AD BC BD CD
Winner B A D

21. Example
Draw a tournament with five players, in which player A defeats everyone, B defeats everyone but A, C is defeated by everyone, and D defeats E.

22. Find all the directed Hamiltonian paths for the following tournaments
Find all the directed Hamiltonian paths for the following tournaments. If there is only one, rank the teams. B A B A D C C D

23. Homework
p (5, 8)