Graphs, Puzzles, & Map Coloring

MATH 102Contemporary Math

S. Rook

Overview

• Section 4.1 in the textbook:– Graph terminology– Graph tracing– More with graphs– Graph coloring

Graph Terminology

Graph Theory

• Very useful to model all sorts of different problems– e.g. most efficient way to travel from point A to

point B (MapQuest), computer networks

• Graph theory is not exclusive to mathematics– Used in other fields such as sociology and political

science– i.e. used to show the connections between

objects

Graphs

• A graph is comprised of a finite set of points called vertices which are connected by one or more lines called edges– Vertices are usually marked with solid dots and labeled– An edge is named by referring to the

two vertices it connects • Adjust naming if two or more edges connect the

same pair of vertices

– What are the edges and vertices of the given graph?

Graphs (Continued)

• Only the vertices and the edges that connect them are important, NOT the shape of a graph!– e.g. Redraw the graph on the previous slide so that it

still illustrates the same relationships• When naming edges in a general graph, order is NOT

important– Order IS important for a directed graph which has

edges that go only in one direction rather than two– e.g. What type of edges would go in only one

direction in the graph of a road map?

Graph Tracing

Graph Tracing

• To trace a graph, we start at a vertex and traverse all edges of the graph ONCE– i.e. No edge can be used twice– Not all graphs can be traced

• A graph is connected if it is possible to start from a vertex and reach any other vertex by following edges

• A bridge is an edge such that if it is removed, the graph is no longer connected

• Is the following graph connected? What are the bridges?

Graph Tracing (Continued)

• An odd vertex has an odd number of edges entering into it; an even vertex has an even number of edges entering into it– e.g. Consider the graph on the previous slide. List the set of

odd vertices and the set of even vertices

• Euler’s Theorem on Graph Tracing: A graph can be traced if:– The graph is connected AND– The graph has zero OR two odd vertices

• If the graph has two odd vertices, the tracing must start with one odd vertex and end at the other

– See pages 141-2 in the textbook for theory

Graph Tracing (Example)

Ex 1: Determine whether the graph can be traced. Explain:

a) b)

More with Graphs

More with Graphs

• Path: a traversal of edges from one vertex to another vertex WITHOUT repeating an edge– The number of edges in a path is called the length

• Euler Path: a path which contains all the edges of a graph– i.e. Graph tracing

• Euler Circuit: an Euler path which starts and ends at the same vertex

• Eulerian Graph: a graph that contains ALL even vertices AND is guaranteed to have an Euler Circuit

More with Graphs (Continued)

• Recall the definition of an Eulerian Graph:– Every vertex must be even

• To Eulerize a graph, we add edges to the graph so that all vertices are even– Can only duplicate pre-existing edges (i.e. cannot

create an edge)

More with Graphs (Example)

Ex 2: Consider the following for each graph:

a) Is the graph traceable?b)Find a path from F to C. What is the length of the

path?c) Eulerize the graph

Graph Coloring

Graph Coloring

• To color a graph, we assign “colors” to each vertex such that no two vertices that are connected with an edge are the same “color”– Can use numbers or symbols besides colors

• Coloring the vertices of a graph is a historical problem in graph theory– See page 147 in the textbook

• The vertices of the same “color” can be used to partition objects into non-conflicting groups

Graph Coloring (Example)

Ex 3: Problem 52 on page 151 of the textbook

Summary

• After studying these slides, you should know how to do the following:– Understand graph terminology– Determine whether a graph is traceable– Eulerize a graph– Use graph coloring to solve problems

• Additional Practice:– See suggested problems in 4.1

• Next Lesson:– Calculating in Other Bases (Section 5.3)