---
tag: 0W1C598HMS
kind: statement
role: theorem
name: Euler's formula for planar graphs
msc: [05C10]
uses: [0FFAYNMAJZ, 029JQEK15B]
sources:
  - url: https://en.wikipedia.org/wiki/Euler_characteristic
    locator: Plane graphs
---

## Statement

Let $G$ be a finite [[connected|0FFAYNMAJZ]] [[planar graph|029JQEK15B]], drawn in the plane with no crossing edges.
Let $v$, $e$ and $f$ be the numbers of vertices, edges and faces of the drawing, where the unbounded face counts.
Then $v - e + f = 2$.

## Remarks

Proof idea: if the graph has a cycle, delete an edge of the cycle. This removes one edge and one face.
At the end you have a [[spanning tree|0FQ5B8S1ZC]], with $e = v - 1$ and $f = 1$.
The same formula holds for the vertices, edges and faces of a convex polyhedron.

## Explanations

### L1 human

Draw a connected graph without crossings and count its vertices, edges and regions.
Vertices minus edges plus regions is always $2$. For a square, this is $4 - 4 + 2$.