=> 10-3x+x=2 (bcz each face act as a region) so get x=8 then they ask no of edge so it will be 3x=24 ans

21 votes

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_______________.

40 votes

Best answer

By Euler formula for connected planar graph,

$n - e + f = 2$

$10 - 3f/2 + f = 2 $ (An edge is part of two faces)

$f/2 = 8$

$f = 16$

$e = 3f/2 = 24$

http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/planarity.htm

1

@Arjun, will questions like this depending on concepts of Planar Graphs (faces etc) will it be asked ? in GATE 2016 ?

0

@Arjun"number of edges on each face is three"

f=3e I am getting

how r u getting e=3f ?

Could u give me a simple example to understand this line?

9

number of edges on each face is three,

max edges = 2e in any graph

here 3 faces have 3 edges each so

3f= 2e

e= 3f/2

0

For every face we need 3 edges. But en edge serves two faces (face - edge - face). So, for f faces we get 3f/2 edges.

1

Can we use the formula :

**e <= 3n-6** for *connected planar graph*

for n=10

**e <= 30-6= 24**

Please correct if wrong approach.

0

Arjun sir, in this question are we talking about only bounded regions??? Because as I know unbounded regions always have degree 0

0

@Arjun sir,

in the question given:

No of edge of every face is three

means f= 3e (wrong)

e = 3 f ( right)

i also know that every edge cover 2 faces.

e= 3f/2

plz check sir ...I am going right or wrong...

33 votes

Gven

no of vertices = 10

The no of edges on each face = 3

Let the no of vertices be 'n' ,no of edges be 'e' , the no of regeions ( faces) be ' r ' .

But

The sum of the degrees of the regions ( faces ) = 2( no of edges)

d(R1) + d(R2) + d(R3) +..............(r times) = 2e

3+3+3+......................(r times )= 2e

3r = 2e

r = 2e/3

By euler's formula we have

n - e + r = 2

10 - e + 2e/3 = 2

10 - e/3 = 2

30 - e = 6

e = 24

The no of edges = 24.

11 votes

By Eulers Formula

N-e+f=2

Where N is the number of Vertices, e is the number of edges and f is the number of face

Here N=10

and it is given that Number of edges on each faces is three , and Each edge is part of two faces..

So N-e+f=2 becomes 10-3*f/2+f=2

=> f=16

Now N-e+f=2 gives e=24

So number of edges in given graph will be 24.

N-e+f=2

Where N is the number of Vertices, e is the number of edges and f is the number of face

Here N=10

and it is given that Number of edges on each faces is three , and Each edge is part of two faces..

So N-e+f=2 becomes 10-3*f/2+f=2

=> f=16

Now N-e+f=2 gives e=24

So number of edges in given graph will be 24.

0 votes

WHOSOEVER Folks drawing the graph like this :

and thinking why the relation "**3R = 2E" **is not satisfying because the open region is not surrounded by 3 vertices.

In Question, it is clearly mention " number of edges on each face is three".

Just to point out, because it took me 30 min to understand Where I'm wrong while drawing this graph.