Hamiltonian Circuits and Paths

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.

Exploration f h d c G e a b i j

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.

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.

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.

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

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.

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.

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

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.

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.

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.

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.

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. Game AB AC AD BC BD CD Winner B A D

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

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

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.

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?

Practice Problems

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

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.

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?

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

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

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?