1. Graphs, Puzzles, & Map Coloring
MATH 102
Contemporary Math
S. Rook

2. Overview Section 4.1 in the textbook: Graph terminology Graph tracing
More with graphs
Graph coloring

3. Graph Terminology

4. 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

5. 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?

6. 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?

7. Graph Tracing

8. 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?

9. 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 in the textbook for theory

10. Graph Tracing (Example)
Ex 1: Determine whether the graph can be traced. Explain: a) b)

11. More with Graphs

12. 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

13. 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)

14. More with Graphs (Example)
Ex 2: Consider the following for each graph:
a) Is the graph traceable?
Find a path from F to C. What is the length of the path?
c) Eulerize the graph

15. Graph Coloring

16. 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

17. Graph Coloring (Example)
Ex 3: Problem 52 on page 151 of the textbook

18. 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)