These linked projects, some of which can stand alone, lead students to discovery certain relationships among the number of vertices, edges, and faces of planar graphs and polyhedra.
The project "Depicting Polyhedra with Schlegel diagrams" can stand alone or augment the other projects. Students explore polyhedra using manipulatives and compare different two-dimensional representations.
"Discovering Euler's Formula for Planar Graphs" has students find the formula from data generated by graphs they draw; it can stand alone or serve as the foundation for either of the following projects.
In "Discovering that K5 is Not Planar" students establish another relationship for a particular family of planar graphs and generalize to an inequality that holds for all planar graphs. Combining this inequality and Euler's formula gives a way to determine that some graphs are not planar, no matter how they are drawn. In particular, K5 is not planar.
"Discovering that K3,3 is Not Planar" parallels the previous project with a different inequality for graphs that contain no triangles of edges. The result is that K3,3 ("the utility graph") is not planar.
"Proofs & Postscript" gives some ideas for the proofs of results that students discover and explains Kuratowski's theorem, a characterization of planar graphs that relies on K5 and K3,3. It also mentions a possible extension and provides references.
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808 and Grant No. ESI-0554309. 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.