aim: graph theory – map coloring course: math literacy do now: what do you notice about this map...

Aim: Graph Theory – Map Coloring Course: Math Literacy

Do Now: What do you notice about this map of South America?

Aim: What does map coloring have to do with Graph theory?

4 colors used

no 2 borderingcountries are the same color.

would 3 colorsalso work?

Four-Color Theorem 1977


Planar Graph Puzzle

Gas Electricity Water

Can each utility run pipes/lines to each hours without letting any pipes/lines cross over each other?

no lights!


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.


The Product Rule

Four-Color theorem proven by converting into a Graph Theory problem.

hypothetical countrieslabeled as letters

letters representvertices

two vertices are connected with an edge if two countries are neighbors

Four-Color Theorem – every planar graph is 4-colorable.


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.

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

Four-color Theorem guarantees four colors at most


Model Problem

Two-colorable, 3-colorable or 4-colorable?


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.


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


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?


Model ProblemSki club

Student Gov.

Debate Club

Honor Society

Student News

Comm. Outreach

Campus Dems

Campus Repubs.

Ski club x x x xStud Gov. x x x xDebate x x x xHonor x x x x xNews x x x xOutreach x x x x xDemocrats x x xRepubs x x x

CR

CD

CO

SN

HS

DC

SG

SCclubs connected by edge cannot meet at same time

a color will correspond to a time slot


Model Problem

CR

CD

CO

SN

HS

DC

SG

SC

clubs connected by edge cannot meet at same time

a color will correspond to a time slot

3 colors - 3 time

slots needed

chromatic number of 3


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.


The Product Rule

aim: graph theory – map coloring course: math literacy do now: what do you notice about this map...

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