Hamiltonian Circuits and Paths

1. Hamiltonian Circuits and Paths

2. Exploration • Let’s pretend that you are a city inspector but this time you must inspect the fire hydrants that are located at each of the street intersections. To optimize your route, you must find a path that begins at the garage, G, visits each intersection exactly once, and returns to the garage.

3. Exploration f h d c G e a b i j

4. One Path Works! • Notice that only one path meets these criteria. • It is path G, h, f, d, c, a, b, e, j, i, G. • Also notice, that is not necessary that every edge of the graph be traversed when visiting each vertex exactly once.

5. Sir William Rowan Hamilton • In the 19th century, an Irishman named Sir William Rowan Hamilton (1805-1865) invented a game called the Icosian game. • The game consisted of a graph in which the vertices represented major cities in Europe.

6. The Icosian Game • The object of the game was to find a path that visited each of the 20 vertices exactly once. • In honor of Hamilton and the his game, a path that uses each vertex of a graph exactly once is known as a Hamiltonian path. • If the path ends at the starting vertex, it is called a Hamiltonian circuit.

7. You Try • Try to find the Hamiltonian circuit in each of the graphs below.

8. Intriguing Results • Mathematicians are intrigued y this type of problem, because a simple test for determining whether a graph has a Hamiltonian circuit has not been found. • The search continues but it now appears that a general solution may be impossible.

9. Hamiltonian Theorem • This theorem guarantees the existence of a Hamilton circuit for certain kinds of graphs. • If a connected graph has n vertices, where n>2 and each vertex has degree of at least n/2, then the graph has a Hamilton circuit.

10. Degrees • Check the degrees of the figures in the graphs below.

11. Finding The Hamiltonian • Since each of the five vertices of the graph has degrees of at least 5/2, the graph has a Hamiltonian circuit. • Unfortunately, the theorem does not tell us how to find the circuit.

12. Hamiltonian Circuits • If a graph has some vertices with degree less than n/2, the theorem does not apply. • The second two of the figures that are drawn have vertices that have a degree less than 5/2, so no conclusion can be drawn. • By inspection, the second figure has a Hamiltonian circuit but the last figure does not.

13. Comparison to Euler Circuits • As with Euler circuits, it often is useful for the edges of the graph to have a direction. • If we consider a competition where every player must play every other player. • This can be shown by drawing a complete graph where the vertices represent the players.

14. Competition Example • In this situation, a directed arrow from vertex A to vertex B would mean that player A defeated player B. • This type of digraph is known as a tournament. • One interesting property of such a digraph is that every tournament contains a Hamilton path which implies that at the end of the tournament it is possible to rank the teams in order, from winner to loser.

15. Example • Suppose Four teams play in the school soccer round robin tournament. The results are as follows: • Draw a digraph to represent the tournament. Find a Hamiltonian path and then rank the participants from winner to loser.

16. Example (cont’d) • Remember that a tournament results from a complete graph when direction is given to the edges. • There is only one Hamiltonian path for this graph, DBAC. • Therefore, D is first, B is second, A is third and C is fourth. B A D C

17. Practice Problems • Apply the Hamiltonian theorem to the graphs below and indicate which have Hamiltonian circuits. Explain why.

18. Practice Problems (cont’d) • Give two examples of situations that could be modeled by a graph in which finding a Hamiltonian path or circuit would be of benefit. • a. Construct a graph that has both an Euler and a Hamiltonian circuit. b. Construct a graph that has neither an Euler now a Hamiltonian circuit.

19. Practice Problems (cont’d) 4. Hamilton’s Icosian game was played on a wooden regular dodecahedron. In the planar representation of the game, find a Hamiltonian circuit for the graph. Is there only one Hamiltonian circuit for the graph? Can the circuit begin at any vertex?

20. Practice Problems

21. Practice Problems (cont’d) • Draw a tournament with five players, in which player A beats everyone, B beats everyone but A, C is beaten by everyone and D beats E. • Find all the directed Hamiltonian paths for the following tournaments: B A B A C D D C

22. Practice Problems (cont’d) • Draw a tournament with 3 vertices in which a. One player wins all of the games it plays. b. Each player wins one game. c. Two players lose all of the games they play. • Draw a tournament with five vertices in which there is a 3-way tie for first place.

23. Practice Problems (cont’d) • When ties exist in a ranking for a tournament, is there a unique Hamiltonian path for the graph? Explain why or why not. • In a tournament a transmitter is a vertex with a positive outdegree and a zero indegree. A receiver is a vertex with a positive indegree and a zero outdegree. Why can a tournament have at most one transmitter and one receiver?

24. Practice Problems (cont’d) • Consider the set of preference schedules: A B C D B C B B D C C D D A A A 8 5 6 7

25. Practice Problems (cont’d) The first preference schedule could be represented by the following tournament: B A D C

26. Practice Problems (cont’d) • Construct tournaments for the other three preference schedules. • Construct a cumulative preference tournament that would show the overall results of the four preference schedules. • Is there a Condorcet winner in the election? • Find a Hamiltonian path for the cumulative tournament. What does this path indicate?