In a previous post, I proved that no 5-connected maximal planar graph is perfect.
My proof, with slight modifications, can show that if a maximal planar graph with minimum degree 5 is perfect, then its connectivity is exactly 3. But does a maximal planar perfect graph with minimum degree 5 and connectivity 3 exist?
Naturally, I wanted to ask whether there exists a 5-connected planar graph that is perfect. In some computer searches, I couldn't even find a planar perfect graph with minimum degree 5.
