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Euler's Gem by David S. Richeson

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CHAPTER 13

PLANAR GRAPHS, GEOBOARDS, AND BRUSSELS SPROUTS

In most sciences one generation tears down what another has built, and what one has established, another undoes. In mathematics alone each generation adds a new story to the old structure.—Hermann Hankel1

In the previous chapter we saw Cauchy’s clever technique for proving Euler’s formula. He took a polyhedron, removed a face, and projected the rest down onto the plane. Then he proved that VE + F = 1 for this figure, so VE + F = 2 for the polyhedron. The connection to graph theory should be obvious. At first glance it appears that it would be trivial to generalize Euler’s formula to graphs that are not projections of polyhedra and which may possess edges that are curved.

The difficulty ...

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