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

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

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.

10 Discrete Mathematical Structures: Theory and Applications 4 Graph Algorithms

10 Discrete Mathematical Structures: Theory and Applications 5 Graph Algorithms

10 Discrete Mathematical Structures: Theory and Applications 6 Graph Algorithms

10 Discrete Mathematical Structures: Theory and Applications 7 Graph Algorithms

10 Discrete Mathematical Structures: Theory and Applications 8

10 Discrete Mathematical Structures: Theory and Applications vknowndvpv v1F00 v2F v3F v4F v5F v6F v7F vknowndvpv v1T00 v2F2v1 v3F v4F1v1 v5F v6F v7F vknowndvpv v1T00 v2F2v1 v3F3v4 T1v1 v5F3v4 v6F9v4 v7F5v vknowndvpv v1T00 v2T2v1 v3F3v4 T1v1 v5F3v4 v6F9v4 v7F5v4 vknowndvpv v1T00 v2T2v1 v3T3v4 T1v1 v5F3v4 v6F8v3 v7F5v4 8 vknowndvpv v1T00 v2T2v1 v3T3v4 T1v1 v5T3v4 v6F8v3 v7F5v4 vknowndvpv v1T00 v2T2v1 v3T3v4 T1v1 v5T3v4 v6F6v7 T5v4 6 vknowndvpv v1T00 v2T2v1 v3T3v4 T1v1 v5T3v4 v6T6v7 T5v4 From 1 to 6

10 Discrete Mathematical Structures: Theory and Applications 10 Graph Algorithms

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

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 Graph G, in Figure , divides the plane into different regions, called the faces of G.

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

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

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

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

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

10 Discrete Mathematical Structures: Theory and Applications 18

10 Discrete Mathematical Structures: Theory and Applications 19

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