Three applications of Euler’s formula

  • Martin Aigner
  • Günter M. Ziegler


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.


Span Tree Plane Graph Great Circle Dual Graph Antipodal Point 
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    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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