Linear Bound Lemma

Theorem
For a simple connected planar graph $G_n$, where $n \geq 3$ is a number of vertices:


 * $m \leq 3 n − 6$, where $m$ is a number of edges.

Proof
Let $f$ denote the number of faces of $G_n$.

Let the sequence $\left\langle{s_i}\right\rangle_{i \mathop = 1}^f$ be the regions of a planar embedding of $G_n$.

Consider the sequence $\left\langle{r_i}\right\rangle_{i \mathop = 1}^f$ where $r_i$ denotes the number of boundary edges for $s_i$.

Since $G$ is simple, then (by the definition of planar embedding):
 * every region has at least $3$ boundary edges
 * every edge is a boundary edge of at most two regions in the planar embedding.

Using this two facts, we can find the boundary for $\displaystyle \sum \limits_{i + 1}^f r_i$ as:


 * $3 f \le \displaystyle \sum \limits_{i + 1}^f r_i \le 2m$

Now calculating the Euler Polyhedron Formula with $f \le 2 m /3$, we will arrive to $m \le 3 n − 6$.