2. Lecture 16: Graph Theory III Discrete Mathematical Structures: Theory and Applications

3. 10 Discrete Mathematical Structures: Theory and Applications 2 Learning Objectives  Learn the basic properties of graph theory  Learn about walks, trails, paths, circuits, and cycles in a graph  Explore how graphs are represented in computer memory  Learn about Euler and Hamilton circuits  Explore various graph algorithms  Examine planar graphs and graph coloring

4. 10 Discrete Mathematical Structures: Theory and Applications 3 Graph Algorithms  Graphs can be used to show how different chemicals are related or to show airline routes. They can also be used to show the highway structure of a city, state, or country.  The edges connecting two vertices can be assigned a nonnegative real number, called the weight of the edge.  If the graph represents a highway structure, the weight can represent the distance between two places, or the travel time from one place to another.  Such graphs are called weighted graphs.

5. 10 Discrete Mathematical Structures: Theory and Applications 4 Graph Algorithms

6. 10 Discrete Mathematical Structures: Theory and Applications 5 Graph Algorithms

7. 10 Discrete Mathematical Structures: Theory and Applications 6 Graph Algorithms

8. 10 Discrete Mathematical Structures: Theory and Applications 7 Graph Algorithms

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11. 10 Discrete Mathematical Structures: Theory and Applications 10 Graph Algorithms

12. 10 Discrete Mathematical Structures: Theory and Applications 11 Planar Graphs and Graph Coloring

13. 10 Discrete Mathematical Structures: Theory and Applications 12 Planar Graphs and Graph Coloring  A graph is a planar graph if and only if it has a pictorial representation in a plane which is a plane graph. This pictorial representation of a planar graph G as a plane graph is called a planar representation of G.  Let G denote the plane graph in Figure 10.111. Graph G, in Figure 10.111, divides the plane into different regions, called the faces of G.

14. 10 Discrete Mathematical Structures: Theory and Applications 13 Planar Graphs and Graph Coloring

15. 10 Discrete Mathematical Structures: Theory and Applications 14 Planar Graphs and Graph Coloring

16. 10 Discrete Mathematical Structures: Theory and Applications 15 Planar Graphs and Graph Coloring

17. 10 Discrete Mathematical Structures: Theory and Applications 16 Planar Graphs and Graph Coloring

18. 10 Discrete Mathematical Structures: Theory and Applications 17 Planar Graphs and Graph Coloring

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21. 10 Discrete Mathematical Structures: Theory and Applications 20 Planar Graphs and Graph Coloring