Lecture 14: Graph Theory I

Lecture 14: Graph Theory I Discrete Mathematical Structures: Theory and Applications

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 • Learn about isomorphism of graphs • Explore various graph algorithms • Examine planar graphs and graph coloring Discrete Mathematical Structures: Theory and Applications

Graph Definitions and Notation • The Königsberg bridge problem is as follows: Starting at one land area, is it possible to walk across all of the bridges exactly once and return to the starting land area? • In 1736, Euler represented the Königsberg bridge problem as a graph, as shown in Figure 10.1(b), and answered the question in the negative. Discrete Mathematical Structures: Theory and Applications

Graph Definitions and Notation • Consider the following problem related to an old children’s game. Using a pencil, can each of the diagrams in Figure 10.2 be traced, satisfying the following conditions? • The tracing must start at point A and come back to point A. • While tracing the figure, the pencil cannot be lifted from the figure. • A line cannot be traced twice. • As in geometry, each of points A, B, and C is called a vertex of the graph and the line joining two vertices is called an edge. Discrete Mathematical Structures: Theory and Applications

Graph Definitions and Notation • Suppose there are three houses, which are to be connected to three services — water, telephone, and electricity — by means of underground pipelines. • The services are to be connected subject to the following condition: The pipes must be laid so that they do not cross each other. Consider three distinct points, A, B, C, as three houses and three other distinct points, W , T, and E, which represent the water source, the telephone connection point, and the electricity connection point. Discrete Mathematical Structures: Theory and Applications

Graph Definitions and Notation • Try to join W , T, and E with each of A, B, and C by drawing lines (they may not be straight lines) so that no two lines intersect each other (see Figure 10.3). • This is known as the three utilities problem. Discrete Mathematical Structures: Theory and Applications

Graph Definitions and Notation Discrete Mathematical Structures: Theory and Applications

Walks, Paths, and Cycles Discrete Mathematical Structures: Theory and Applications

Lecture 14: Graph Theory I