Linear Bound Lemma

Theorem
Let $G_n$ be a simple connected planar graph with $n$ vertices.

Then:
 * $m \le 3 n − 6$

where $m$ is the number of edges.

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

Let $\sequence {s_i}_{i \mathop = 1}^f$ be a sequence of regions of a planar embedding of $G_n$.

Consider the sequence $\sequence {r_i}_{i \mathop = 1}^f$ where $r_i$ denotes the number of boundary edges of $s_i$.

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 of $\ds \sum_{i \mathop = 1}^f r_i$ as:


 * $3 f \le \ds \sum_{i \mathop = 1}^f r_i \le 2m$

Now, as $f \le \dfrac 2 3 m$: