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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.

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The icosahedral graph (the graph of the vertices and edges of an icosahedron) is a 5-vertex-connected, and 5-edge-connected, planar graph. It has 12 vertices and 30 edges. Due to the bounds you mention in the question, it is the smallest possible example.

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