By Euler formula for connected planar graph,
$n - e + f = 2$
$10 - 3f/2 + f = 2 $ (An edge is part of two faces)
$f/2 = 8$
$f = 16$
$e = 3f/2 = 24$
This is a "planner" graph with 10 vertices and each face has only 3 edges.
Answer is same that is 24.
@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?
number of edges on each face is three,
max edges = 2e in any graph
here 3 faces have 3 edges each so
The graph looks like this.
Can we use the formula :
e <= 3n-6 for connected planar graph
e <= 30-6= 24
Please correct if wrong approach.
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 ' .
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.
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.