Number of regions in the planar embedding of a connected, $3$-regular graph with $24$ vertices

Question

Let $G$ be a connected, $3$-regular, and planar graph with $24$ vertices. How many regions are in a planar embedding of $G$?

Any help to solve this problem would be highly appreciated. Thanks in advance!

Answer 1

By Euler's formula, we know that

$V-E+F=2$

For $K_{2,3}$, $V=5$, $E=6$. Hence,

$5-6+F=2$, which means $F=3$

Now $G$ is $3$-regular. Hence $d(v) = 3 \\ \forall v \in V$

Now $2|E| = \sum_{v \in V}d(v) =3 \times24 \implies |E|=36$

Here, $E=36, V=24$

By Euler's formula, $F=2-V+E=2-24+36=14$