Exploring Vertex and Map Coloring in Math Activities
This overview discusses vertex coloring, an essential concept in graph theory, and its applications in scheduling, such as creating exam timetables without student conflicts. By representing classes as vertices and shared students as edges, we can find the minimum number of periods required. Additionally, map coloring rules ensure adjacent areas do not share colors, emphasizing efficiency through examples like the Four-Color Theorem, proven by Appel and Haken in 1976. Other applications include scheduling meetings, organizing guests, and managing resources.
Exploring Vertex and Map Coloring in Math Activities
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Presentation Transcript
“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?
Vertex Coloring • Rule: Two adjacent vertices cannot be the same color • Goal: Use the minimum number of colors to color all vertices A B E C D
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.
A Possible Solution A B C E F G H D
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.
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.
