If G is a simple 4-regular planar graph, is the maximum degree of a face 4?

Question

Are there any restrictions in general for Simple k-regular planar graphs?

Also is it possible for a face to have degree 1 or degree 2 in this case? I assume that it could be possible for an isolated point but then the graph wouldn't be k-regular.

Answer 1

In a simple graph, a face can never have degree $1$ or $2$. (Degree-$1$ faces correspond to loops, and degree-$2$ faces correspond to multiple edges between the same pair of vertices. Neither of these is allowed in a simple graph.)

However, a face can have degree more than $4$, and this remains the case if the graph is $4$-regular. For example, consider the skeleton graph of the pentagonal antiprism. This is $4$-regular with $10$ vertices; it has $10$ faces of degree $3$ and $2$ faces of degree $5$. Here is a plane embedding:

Other antiprisms give us examples of $4$-regular planar graphs with faces that have arbitrarily many sides.