# What is the smallest $5$-vertex-connected ($5$-edge-connected) planar graph?

A planar graph cannot be $6$-connected because the number of edges of a planar graph with $n$ vertices is at most $3n-6$, while a $6$-connected graph with $n$ vertices must have at least $3n$ edges.

• Are there $5$-vertex-connected planar graphs, and if yes, what is the smallest ?
• Is there an even smaller example for a $5$-edge-connected planar graph ?

Since a $5$-connected graph with n vertices must have at least $\frac{5n}{2}$ edges, the condition is $\frac{5n}{2}\le 3n-6$, which implies $n\ge 12$, so a $5$-connected planar graph must have at least $12$ vertices.