A graph is planar if it can be drawn in the plane ℝ2 without crossing edges (or, equivalently, on the 2-dimensional sphere S2). We talk of a plane graph if such a drawing is already given and fixed. Any such drawing decomposes the plane or sphere into a finite number of connected regions, including the outer (unbounded) region, which are referred to as faces. Euler’s formula exhibits a beautiful relation between the number of vertices, edges and faces that is valid for any plane graph. Euler mentioned this result for the first time in a letter to his friend Goldbach in 1750, but he did not have a complete proof at the time. Among the many proofs of Euler’s formula, we present a pretty and “self-dual” one that gets by without induction. It can be traced back to von Staudt’s book “Geometrie der Lage” from 1847.
CITATION STYLE
Aigner, M., & Ziegler, G. M. (2018). Three applications of Euler’s formula. In Proofs from THE BOOK (pp. 89–94). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-57265-8_13
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