Minimum Degree Bound for Simple Planar Graph

Theorem
Let $G$ be a simple connected planar graph.

Then:


 * $\map \delta G \le 5$

where $\map \delta G$ denotes the minimum degree of vertices of $G$.

Proof
$G$ is a simple connected planar graph with $\map \delta G \ge 6$.

Let $m$ and $n$ denote the number of edges and vertices respectively in $G$.

Then by the Handshake Lemma:

This contradicts the Linear Bound Lemma:
 * $m \le 3 n - 6$

Hence $\map \delta G \le 5$.