“Coloring in Math Class” Julie March Tracey Clancy Onondaga Community College

Overview Vertex Coloring –Activity –Other applications Map Coloring –Activities

Question How do we make an exam schedule without any conflicts? AstronomyBiologyChemistry Differential EnglishFrenchGermanHistory KeplerDarwinBohrL'HôpitalHamiltonVoltaireEulerLincoln CopernicusPasteurLavoisierEulerDarwinL'HôpitalBohrVoltaire HerschelHarveyPlankHamiltonHerschelPasteurKeplerEuler LagrangeVesaliusLaPlaceEuclidThomsonLavoisierMendelCopernicus

Vertex Coloring Rule: Two adjacent vertices cannot be the same color Goal: Use the minimum number of colors to color all vertices D A E B 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. AstronomyBiologyChemistry Differential EnglishFrenchGermanHistory KeplerDarwinBohrL'HôpitalHamiltonVoltaireEulerLincoln CopernicusPasteurLavoisierEulerDarwinL'HôpitalBohrVoltaire HerschelHarveyPlankHamiltonHerschelPasteurKeplerEuler LagrangeVesaliusLaPlaceEuclidThomsonLavoisierMendelCopernicus

Work At Seats

A Possible Solution ACB FEG 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

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 ?