# What is the maximum connectivity of a planar graph?

The Icosahedral Graph is a simple 5-connected planar graph. Is there a 6-connected planar graph?

In general, is there a theoretical maximum on the vertex connectivity of planar graphs? This is spurred since the number of edges is bounded by $$3n-6$$, and that could set an upper bound on the vertex connectivity.

• A planar graph must have a vertex of degree less than six. – Gerry Myerson Dec 4 '18 at 22:57

I think you pointed at the exact right direction. As the number of edges is at most $$3n-6$$, there must be a vertex $$v$$ of degree at most $$5$$. Delete the neighbors of $$v$$, and the graph falls apart (unless it had at most 6 vertices).