Hamilton Paths & Circuits in Special Graphs
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Transcript of Hamilton Paths & Circuits in Special Graphs
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By: Todd Waters & Maya Robinson
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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.
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A Petersen graph is a graph with 10 vertices and 15 edges.
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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.
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A Petersen graph has a Hamilton path but no Hamiltonian cycle. In other words a Peterson graph does not have a Hamilton Circuit.
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Definition Example
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.
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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.
A B
C D E
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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.
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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
A
B
C
D
E
F
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Regular Dodecahedron A dodecahedron is a regular polyhedron with twelve flat faces each a regular pentagon with threemeeting at each vertex.
Regular IcosahedronAn icosahedraon is a regularpolyhedron with 20 identicalequilateral triangular faces with
eachmeeting at each vertex.
Dodecahedron
Icosahedron
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Has a Hamilton circuit Order of travel:
◦ 1-5-6-15-14-13-12-11-10-9-18-19-20-16-17-7-8-4-3-2-1
◦ 2-3-4-5-1-14-15-6-7-8-9-10-11-19-18-17-16-20-13-12-2 None of the vertices were hit twice
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Has a Hamilton circuit Order of travel:
◦ 3-2-4-5-6-11-10-9-8-12-7-1-3
◦ 7-8-3-9-4-2-5-10-12-11-6-1-7
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Hamilton path that starts at I◦ Is it possible: YES!◦ I-B1-C1-B4-C4-B3-C3-B2-C2
A path was made where each vertex in the graph was visited only once.
C1
B4
B1
C4
C2
C3
B2I
B3
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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
C1
B4
B1
C4
C2
C3
B2I
B3
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Hamilton path that starts at one of the end corners vertices and ends at I◦ Is it possible: YES!
C4-B3-C3-B2-C2-B1-C1-B4-I A path was made where each vertex in
the graph was visited only once
C1
B4
B1
C4
C2
C3
B2I
B3
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Hamilton path that starts at one of the corner vertices and ends at one of the boundary vertices.◦ Is it possible?: NO!
The interior vertex (I) creates a problems
C1
B4
B1
C4
C2
C3
B2I
B3
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Hamilton circuit:◦ Is it possible: NO!
The interior vertex(I) creates a problem. It is now impossible for
us to visit each vertex once.
C1
B4
B1
C4
C2
C3
B2I
B3
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Terry, E., Class Notes, July 2010.
Wikipedia, Internet, July 2010.