Question 64

1. Given:

   • Connected planar graph $G$ with $V = 10$ vertices.
   • Every face is bounded by exactly 3 edges: $\text{deg}(f) = 3$ for each face $f$.

2. Handshaking Lemma for Faces:
   Each edge is shared by two faces (or bounds two sides of a face), so:

   $$\sum_{f \in F} \text{deg}(f) = 2E \implies 3F = 2E \implies F = \frac{2E}{3}.$$

3. Euler's Formula:
   For any connected planar graph:

   $$V - E + F = 2.$$

   Substitute $V = 10$ and $F = \frac{2E}{3}$:

   $$10 - E + \frac{2E}{3} = 2$$
   $$10 - \frac{E}{3} = 2$$
   $$\frac{E}{3} = 8 \implies E = 24.$$

The number of edges in $G$ is $24$.