Minimum Degree Bound for Simple Planar Graph

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


 * $\delta \left(G \right) \leq 5$, where $\delta$ is a minimum degree of a graph.

Proof
Proving by contradiction. Consider the counter hypothesis:


 * $G$ is a simple planar graph and $\delta \left(G \right) \geq 6$.

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

Then, by the Handshake Lemma:


 * $m \geq 3n$

That contradicts the Linear Bound.