1. Graph Theory
Unit 2

2. 2.12 Chromatic Number

3. Definition
The Chromatic Number of a graph is the least number of colours required to colour all of the vertices of the graph, such that no two adjacent vertices are the same colour.
The purpose:
to determine the number of colours required to colour a map
to categorize information into as few groups as possible
Categorize/Group data based on INCOMPATIBILITIES

4. A Map How many countries? 13 Draw that many vertices
What are you categorizing?
Countries that share borders
Edges represent the relationship:
“share a border with”
Connect the appropriate vertices
Goal?
Least number of colours

5. A Map to A Graph V G S FG C E P Bo A Pa U B

6. 4 Colours A Graph to An Answer V G S C E P A U B V G S C B E P U A CFG C E P Bo A Pa U B V G S FG C B E P Bo Pa U A C
4 Colours

7. A Chromatic Situation You are the Team Leader for a major company.
You know there are certain personalities that cannot work together.
You have a team of 5 people and you want to make as few groups as possible, understanding who is compatible
You know:
Beth cannot work with Sue
David and Jonas will never get any work done
Sue and Stanley are unproductive
David and Sue are poorly matched
Stanley and David are not compatible
How many groups do you need?

8. Process 1) You must organize your information:
5 employees
Who is ‘incompatible’
Beth cannot work with Sue
Sue and Stanley are unproductive
David and Sue are poorly matched
Stanley and David are not compatible
David and Jonas will never get any work done
Vertices: Beth, Sue, David, Jonas, Stanley
Edges B Su D St J

9. 3 Groups Process 2) Analyse information to make groups:
Beth cannot work with Sue
Sue and Stanley are unproductive
David and Sue are poorly matched
Stanley and David are not compatible
David and Jonas will never get any work done
Must be in different groups (colours) B B Su Su D D St St J J
3 Groups

10. Chromatic Number Process
Steps:
Colour the vertex of the highest degree
Colour the vertices adjacent to the vertex of highest degree with another colour without given any two adjacent vertices the same colour.
Continue to colour adjacent vertices, REPEAT previously used colours as soon as possible.
**Highest Degree: most edges touching it
**Adjacent: two vertices that are connected by an edge

11. 2.12 Practice
Visions Book 2
Page 52
#1, 11, 12, 14, 15, 18

12. 2.12 Practice 2
Carousel
Page 130
#1-7