Let be a connected planar graph with and . Let be the number of faces determined by the planar embedding1. Then we conclude that

Imporant: Euler’s Theorem can be only used for proving some graphs are non-planar. Cannot be used to prove a graph is planar.

Corollary 1

Let be a planar graph with components. Then in any planar embedding of the number of regions equals:

corollary 2 drawing

Corollary 2

Let is a planar graph with . Then

Corollary 3

Let is a planar bipartite graph with , then


  1. If the graph isn’t presented in its planar embedding, the theorem doesn’t hold.