Number of triangles in a planar graph

Question

Assume that G = (V, E) is a planar graph. If G has 8-vertices and 13 edges then what can be a minimal possible number of triangular regions? What can be a maximal possible number of triangular regions?

Answer 1

The Euler formula tells us that $G$ has $7$ faces. Let $T$ be the number of triangles, therefore each of the remaining $7-T$ faces is not a triangle.

We now do the standard counting of edges by faces. We have $T$ faces with $3$ edges and $7-T$ faces with $4$ or more faces. Therefore our count is at least $3T+4(7-T)$. As usual, we next realize that we counted each edge twice. So our count is $2*13$.

Therefore $$2*13 \geq 3T+4(7-T)=28-T$$

This proves that $T \geq 2$.

The graph must have at least $2$ triangles.

Now, as Perry Elliott-Iverson pointed, $7$ is not possible (as the dual graph would have an odd number of odd vertices). So $2 \leq T \leq 6$. All those can be shown possible.