Editing Euler characteristic (section)

===Plane graphs===
{{See also|Planar graph#Euler's formula}}
The Euler characteristic can be defined for [[Connectivity (graph theory)|connected]] [[plane graph]]s by the same <math>\ V - E + F\ </math> formula as for polyhedral surfaces, where {{mvar|F}} is the number of faces in the graph, including the exterior face.

The Euler characteristic of any plane connected graph {{mvar|G}} is 2. This is easily proved by induction on the number of faces determined by {{mvar|G}}, starting with a tree as the base case. For [[Tree_(graph_theory)|trees]], <math>\ E = V - 1\ </math> and <math>\ F = 1 ~.</math> If {{mvar|G}} has {{mvar|C}} components (disconnected graphs), the same argument by induction on {{mvar|F}} shows that <math>\ V - E + F - C = 1 ~.</math> One of the few graph theory papers of Cauchy also proves this result.

Via [[stereographic projection]] the plane maps to the 2-sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic&nbsp;2. This viewpoint is implicit in Cauchy's proof of Euler's formula given below.