Maximal number of faces

In a planar graph on \(n\) vertices:

Variant
Determine the maximal number of inner faces.
Solution
If a face is not triangular, it can be divided by adding additional edges to two or more faces.

According to Euler’s formula, the graph now has \( 3n-6 \) edges, from which it is possible to create \( \frac{2}3 (3n-6) \) triangular faces, one of which is external.
Answer
A given planar graph can have at most \( 2n-5 \) of inner faces.
Variant
The same problem with the additional condition that the outer face of the graph is bounded by a cycle of length \( k \).
Solution
We complete any graph with the maximum number of inner faces to a triangulation by adding \( k-3 \) edges inside the outer face. This step yields \( k-3 \) extra triangular faces. The resulting graph has a total of \( 2n-4 \) faces, out of which \( 2n-k-8 \) correspond to the original inner faces.
Answer
A given planar graph can have at most \( 2n-k-8 \) of inner faces.