## Exploring Discrete Mathematics Summary

### Monday - Friday, June 26 - June 30, 2006

Day 1: Today we started with Brian leading a discussion of the basic ideas of topology. We did an activity to discover Euler's Formula, which expresses the relationship between faces, edges, and vertices of graphs. We also discussed the four-color problem.

Day 2: For the first hour, we joined the undergraduate faculty for Professor Colin Adams' presentation on knot theory. After that we looked at methods for map-coloring. We colored the western half of the United States trying different methods, and then we looked at an algorithm that will frequently produce a well-colored map. We also looked at Sudoku puzzles and discussed some of the math questions that could arise from them.

Day 3: Today we started by exploring triangular graphs, which are graphs in which each face had exactly three edges. From this we came up with a way to test whether graphs were planar by using an inequality relating vertices and edges. We also discussed complete graphs and bipartite graphs. We then began to discuss what we would be doing for our project.

Day 4: Today we began by discussing the utility problem, which poses the question of whether it is possible for three utilities, gas, electric, and water to each be connected to three houses with no utility lines crossing each other. We proved that it is impossible. We then formalized ideas for our projects. We are going to make a series of self-contained lessons that deal with such ideas as polyhedrons and Schlegel diagrams, Euler's formula, planarity of graphs, and the utility problem. We then paired up, with one group working on an activity having students build polyhedrons and create Schlegel diagrams, and the other group working on an activity to study Euler's formula.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.