Question

Suppose that a connected planar simple graph has 20 vertices, each of degree3. Into how many regions does a representation of this planar graph split the plane?

Answer 1

V= 20

using handshaking lemma E = 30, now,

using V-E+R = 2,

R = 12

Answer 2

Answer:

Answer is 12 regions

Step-by-step explanation:

When a simple planar graph is drawn without edges crossing, the edges and vertices of the graph divide the plane into regions.

We will call each region a face.

Euler's formula for planar graphs is  

Any connected planar graph with v vertices, e edges, and f faces, we have

                            $v - e +f =2$  

The relationship between edges and degrees of vertices is given by

                       $\sum^n_i {deg(v_{i} )} \,= 2q$

They have given that a connected planar simple graph has 20 vertices, each of degree 3.

Therefore the total number of degrees = 20 × 3 = 60

The total number of edges is given by

$60 = 2q$

q = 30

The total number of regions of a connected planar simple graph is given by

$v - e +f =2$

20 - 30 + f  = 2

f = 2 + 10

f = 12

Therefore the 12 regions split the plane of the planar graph.