Minimum Degree Bound for Simple Planar Graph

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $G$ be a simple connected planar graph.

Then:

$\map \delta G \le 5$

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


Proof

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.

$\blacksquare$