hamilton paths & circuits in special graphs

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.

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.

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

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.

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

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

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

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

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

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

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

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

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

Terry, E., Class Notes, July 2010.

Wikipedia, Internet, July 2010.