“Coloring in Math Class”

Julie March

Tracey Clancy

Onondaga Community College

Overview

• Vertex Coloring

–Activity

–Other applications

• Map Coloring

–Activities

QuestionHow do we make an exam schedule without

any conflicts?

Astronomy Biology Chemistry Differential English French German History

Kepler Darwin Bohr L'Hôpital Hamilton Voltaire Euler Lincoln

Copernicus Pasteur Lavoisier Euler Darwin L'Hôpital Bohr Voltaire

Herschel Harvey Plank Hamilton Herschel Pasteur Kepler Euler

Lagrange Vesalius LaPlace Euclid Thomson Lavoisier Mendel Copernicus

Vertex Coloring

• Rule: Two adjacent vertices cannot be the same color

• Goal: Use the minimum number of colors to color all vertices

D

A

EB

C

Scheduling Final Exams• What is the minimum number of exam

periods required, if we do not allow for any student conflicts?

• Create a model letting the vertices represent the classes and using an edge to represent classes that share a student.

Astronomy Biology Chemistry Differential English French German History

Kepler Darwin Bohr L'Hôpital Hamilton Voltaire Euler Lincoln

Copernicus Pasteur Lavoisier Euler Darwin L'Hôpital Bohr Voltaire

Herschel Harvey Plank Hamilton Herschel Pasteur Kepler Euler

Lagrange Vesalius LaPlace Euclid Thomson Lavoisier Mendel Copernicus

Work At Seats

A Possible SolutionA CB

FE G

D H

Other Applications• Scheduling meetings

• Animal habitats

• Fish in tanks

• Table settings for guests at a party

• Traffic patterns at intersections

• Radio frequencies

• Kids in vans for a trip

• Scheduling tasks that require specific processors

A Different Way Color

• Map Coloring:• Rules:

– Areas that share a border cannot have the same color.

– The intention is to use the least number of colors as possible.

Creating a Map Without Lifting Pencil

Creating a Map Without Lifting Pencil

How Many Colors Are Required?

Stained Glass Window Activity

What is the minimum number of colors required?

Four-Color Theorem• In 1853 Francis Guthrie first conjectured that all

maps can be colored with using four colors or less.• In later years, several individuals attempted to

prove this conjecture.• In 1976, after more than a hundred years,

Guthrie’s conjecture was finally proved by Kenneth Appel and Wolfgang Haken. This proof took 500 pages and 1000 hours of computing time.

• This was later known as the four-color theorem.

Questions ?