Hamiltonian Circuits and Paths

1
Hamiltonian Circuits and Paths
2
Exploration

• Lets 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.

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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.

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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.

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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.

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You Try

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

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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.

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Hamiltonian Theorem

• This theorem guarantees the existence of a
  Hamilton circuit for certain kinds of graphs.
• If a connected graph has n vertices, where ngt2
  and each vertex has degree of at least n/2, then
  the graph has a Hamilton circuit.

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Degrees

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

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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.

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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.

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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.

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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.

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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 B D D
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Example (contd)

• 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
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Practice Problems

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

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Practice Problems (contd)

• 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.

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Practice Problems (contd)

• 4. Hamiltons 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?

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Practice Problems
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Practice Problems (contd)

1. 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.
2. Find all the directed Hamiltonian paths for the
   following tournaments

B
A
B
A
C
D
D
C
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Practice Problems (contd)

• 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.

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Practice Problems (contd)

1. When ties exist in a ranking for a tournament, is
   there a unique Hamiltonian path for the graph?
   Explain why or why not.
2. 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?

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Practice Problems (contd)

1. 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
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Practice Problems (contd)

• The first preference schedule could be
  represented by the following tournament

B
A
D
C
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Practice Problems (contd)

1. Construct tournaments for the other three
   preference schedules.
2. Construct a cumulative preference tournament that
   would show the overall results of the four
   preference schedules.
3. Is there a Condorcet winner in the election?
4. Find a Hamiltonian path for the cumulative
   tournament. What does this path indicate?