Question

Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is

• A. 24

Answer 1

The correct option is A 24
Number of vertices (n) = 10
$d\left(r_1\right)=3$

Number of edges (e) = ?
r = e - v + 2

(euler's formula for connected planar graphs)
r = e - 10 + 2 = e - 8 ...(1)

Since every region is bounded by exactly 3 edges and since every edge is exactly double counted, we have the equation

$e=\frac{3r}{2}\Rightarrow r=\frac{2e}{3}$

Substituting this in equation (1) we get,
$\frac{2e}{3}=e-8$

$2e=3e-24\Rightarrow e=24$