Download Graph Theory IV

By: Todd Waters & Maya Robinson



The objective is to visit a number of cities
once and return home with the minimum
amount of travel.
Used by mathematicians, statisticians, and
computer scientist to solve optimization
problems
This relates to our presentation because we
are going to use Hamilton circuits.

A Petersen graph is
a graph with 10
vertices and 15
edges.

A Hamiltonian cycle (or Hamiltonian circuit) is
a cycle in an undirected graph which visits
each vertex exactly once and also returns to
the starting vertex.
•A
path visits each vertex of a
graph once and only once.
A Petersen graph has a
Hamilton path but no
Hamiltonian cycle. In other
words a Peterson graph does
not have a Hamilton Circuit.

A bipartite graph is a
graph whose vertices
can be divided into two
sets X and Y such that
every edge connects a
vertex in X to one in Y;
Which makes, X and Y
independent sets.
Definition
Example

A
C
B
D
E
This example has
no Hamilton circuit
because in a
Hamilton circuit
you must visit each
vertex of the graph
exactly once and
return to the
starting point. In
this example you
cannot do so.

Also, this example
has no Hamilton
circuit because you
must visit each
vertex of the graph
exactly once and
return to the
starting point. In
this example you
cannot do so.

A
D
B
E
C
F

In this example, you have several
Hamilton Circuits. This graph
differs from the other examples
because the independent sets have
an equal number of points.
Hamilton Circuits
◦
◦
◦
◦
◦
◦
A-F-C-E-B-D-A
D-B-F-C-E-A-D
B-D-A-F-C-E-B
C-E-A-D-B-F-C
E-A-D-B-F-C-E
F-B-D-A-E-C-F


Regular Dodecahedron
A dodecahedron is a regular
polyhedron with twelve flat faces
each a regular pentagon with three
meeting at each vertex.
Regular Icosahedron
An icosahedraon is a regular
polyhedron with 20 identical
equilateral triangular faces with each
meeting at each vertex.
Dodecahedron
Icosahedron


Has a Hamilton circuit
Order of travel:
◦ 1-5-6-15-14-13-12-1110-9-18-19-20-16-17-78-4-3-2-1
◦ 2-3-4-5-1-14-15-6-7-89-10-11-19-18-17-1620-13-12-2
 None of the vertices were hit twice


Has a Hamilton circuit
Order of travel:
◦ 3-2-4-5-6-11-10-9-812-7-1-3
◦ 7-8-3-9-4-2-5-10-1211-6-1-7
C1
B1
C2

Hamilton path that
starts at I
◦ Is it possible: YES!
◦ I-B1-C1-B4-C4-B3-C3-B2-C2
I
B4
B2
C4
C3
B3
 A path was made where each vertex
in the graph was visited only once.
C1

Hamilton path that starts at
one of the corner vertices
and end at a different corner
vertex
◦ Is it possible: YES!
 C1-B1-I-B4-C4-B3-C3-B2-C2
 A path was made where each vertex in
the graph was visited only once
B1
C2
I
B4
B2
C4
C3
B3

C1
B1
C2
◦ Is it possible: YES!
I
B2
B4
Hamilton path that
starts at one of the end
corners vertices and
ends at I
 C4-B3-C3-B2-C2-B1-C1-B4I

C3
C4
B3
A path was made where each vertex in the
graph was visited only once

C1
B1
C2
Hamilton path that starts
at one of the corner
vertices and ends at one
of the boundary vertices.
◦ Is it possible?: NO!
I
B4
B2
C4
C3
B3
 The interior vertex (I) creates a
problems

C1
B1
C2
Hamilton circuit:
◦ Is it possible: NO!
 The interior vertex(I)
creates a problem.
I
B4
B2
C4
C3
B3
 It is now impossible for
us to visit each vertex
once.
 Terry,
E., Class Notes, July 2010.
 Wikipedia,
Internet, July 2010.