Hamilton Paths & Circuits in Special Graphs

1. Hamilton Paths & Circuits in Special Graphs By: Todd Waters & Maya Robinson

2. Traveling Salesman Problem • 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.

3. Peterson Graph • A Petersen graph is a graph with 10 vertices and 15 edges.

4. Hamilton Circuit • 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. Hamilton Path • A path visits each vertex of a graph once and only once.

5. Does a Peterson Graph Have a Hamilton Circuit? A Petersen graph has a Hamilton path but no Hamiltonian cycle. In other words a Peterson graph does not have a Hamilton Circuit.

6. Bipartite Graphs • 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

7. Bipartite Graph Ex.1 • 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

8. Bipartite Graph Ex.2 • 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.

9. Bipartite Graph Ex.3 • 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 D B E C F

10. Two Platonic Solids: Dodecahedron and Icosahedron • 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 icosahedraonis a regular • polyhedron with 20 identical • equilateral triangular faces with each • meeting at each vertex. Dodecahedron Icosahedron

11. Dodecahedral Graph • 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

12. Icosahedral Graph • 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

13. A 2 by 2 Grid • 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. B1 C1 C2 I B2 B4 C3 C4 B3

14. A 2 by 2 Grid • 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 C1 C2 I B2 B4 C3 C4 B3

15. A 2 by 2 Grid • 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 B1 C1 C2 I B2 B4 C3 C4 B3

16. A 2 by 2 Grid • 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 B1 C1 C2 I B2 B4 C3 C4 B3

17. A 2 by 2 Grid • Hamilton circuit: • Is it possible: NO! • The interior vertex(I) creates a problem. • It is now impossible for us to visit each vertex once. B1 C1 C2 I B2 B4 C3 C4 B3

18. REFERENCES • Terry, E., Class Notes, July 2010. • Wikipedia, Internet, July 2010.