Is there a cubic (i.e. \( 3 \)-regular) planar graph that contains:
Just 12 hexagonal faces (and no other)?
No, the dual of this graph would be a 6-regular planar graph, and no such exists.
There is no such graph.
Just 12 pentagonal faces (and no other)?
Yes, there is such a graph, it is a graph of one of the Platonic solids – dodecahedron.
One pentagonal face and ten pentagonal faces (and no other)?
From the prescribed number of faces, the searched graph should have a total of \( \frac12 (20 + 10 \ cdot5) = 35 \) edges.
A cubic graph has an even number of vertices and a total of \( \frac32 | V_G | \) edges, so the number of edges is always a multiple of three.
Since 35 is not divisible by three, the graph we are looking for does not exist.