Minimum Degree Bound for Simple Planar Graph

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Let $G$ be a simple connected planar graph.


$\map \delta G \le 5$

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


Aiming for a contradiction, suppose:

$G$ is a simple planar graph and $\map \delta G \ge 6$.

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

Then, by the Handshake Lemma:

$m \ge 3 n$

That contradicts the Linear Bound Lemma.