1. Aim: What does map coloring have to do with Graph theory?
Do Now: What do you notice about this map of South America?
4 colors used
no 2 bordering
countries are the
same color.
would 3 colors
also work?
Four-Color
Theorem 1977

2. no lights! Planar Graph Puzzle
Can each utility run pipes/lines to each hours without letting any pipes/lines cross over each other?
Gas
Electricity
Water
no lights!

3. Planar Graph
planar graph – a graph that can be drawn so that no edges intersect each other (except vertices).
Gas
Electric
Water
House A
House B
House C
Not a planar graph
planar graph
For a graph to be planar it’s only required that the graph can be drawn in at least one way in which the edges do not intersect.

4. The Product Rule
Four-Color theorem proven by converting into a Graph Theory problem.
hypothetical countries
labeled as letters
letters represent
vertices
two vertices are connected with an edge if two countries are neighbors
Four-Color Theorem – every planar graph is 4-colorable.

5. Four-color Theorem guarantees four colors at most
Model Problem
The fictional map shows the boundaries of a rectangular continent. Represent the map as a graph, and then find a coloring of the graph using the least number of colors.
Four-color Theorem guarantees four colors at most
To color – pick a vertex, give it a color and then assign colors to the connected vertices one by one. Try to reuse colors as often as possible

6. Model Problem
Two-colorable, 3-colorable or 4-colorable?

7. Chromatic Number of Graph
Four-Color Theorem – every planar graph is 4-colorable.
non-planar graphs may require many more colors.
Chromatic number of a graph – the smallest number of colors needed to color a graph so that no edge connects vertices of the same color.
Two is the smallest number
2-Colorable Graph Theorem – A graph is
2-colorable if and only if it has no circuits
that consist of an odd number of vertices.

8. Chromatic Number of Graph
2-Colorable Graph Theorem – A graph is
2-colorable if and only if it has no circuits
that consist of an odd number of vertices. G E W A B C

9. Applications of Graph Coloring
Determining the chromatic number of a graph and finding a corresponding coloring of a graph can solve practical problems.
Ex. Eight different school clubs want to schedule meetings on the last day of the semester. Some club members, however, belong to more than one club, so clubs that share members cannot meet at the same time. How many different time slots are required so that all members can attend all meeting?

10. clubs connected by edge cannot meet at same time
Model Problem
Ski club
Student Gov.
Debate Club
Honor Society
Student News
Comm. Outreach
Campus Dems
Campus Repubs. x
Stud Gov.
Debate
Honor
News
Outreach
Democrats
Repubs
clubs connected by edge cannot meet at same time
a color will correspond to a time slot

11. 3 colors - 3 time slots needed
Model Problem
clubs connected by edge cannot meet at same time
a color will correspond to a time slot
3 colors time slots needed
chromatic number of 3

12. Model Problem
Six friends are taking a film history course and, because they procrastinated, need to view several films the night before the final exam. They have rented a copy of each film on DVD, and they have a total of 3 DVD player in different dorm rooms. If each film is 2 hours long and they start watching at 8pm, how soon can they all be finished watching the required films? Make a viewing schedule for the friends.
Film A needs to be viewed by Brian, Chris and Damon Film B needs to be viewed by Allison and Fernando Film C needs to be viewed by Damon, Erin, and Fernand Film D needs to be viewed by Brian and Erin Film E needs to be viewed by Brian, Chris, and Erin.

13. The Product Rule