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Three applications of Euler’s formula

  • Martin Aigner
  • Günter M. Ziegler

Abstract

A graph is planar if it can be drawn in the plane ℝ2 without crossing edges (or, equivalently, on the 2-dimensional sphere S 2). 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.

Keywords

Span Tree Plane Graph Great Circle Dual Graph Antipodal Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    G. D. Chakerian: Sylvester’s problem on collinear points and a relative Amer. Math. Monthly 77 (1970), 164–167.MathSciNetMATHCrossRefGoogle Scholar
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    G. Pick: Geometrisches zur Zahlenlehre, Sitzungsberichte Lotos (Prag) Natur-med. Verein für Böhmen 19 (1899), 311–319.Google Scholar
  3. [3]
    K. G. C. von Staudt: Geometrie der Lage, Verlag der Fr. Korn’schei Buchhandlung, Nürnberg 1847.Google Scholar
  4. [4]
    N. E. Steenrod: Solution 4065/Editorial Note, Amer. Math. Monthly 51 (1944), 170–171.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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