For PLANAR GRAPHS There are........
vertex colouring for vertices
map colouring for regions
and
edge colouring for edges
IN THE QUESTION planar graph requires ( MINIMUM ) vertex colouring .
1)
DRAW A GRAPH WITH 2 VERTICES AND ONE EDGE
( ie., v=2 , e = 1 , f =1 ) IS A PLANAR GRAPH .
AND IT SATISFIES EULER FORMULA .( v - e + f = 2 )
The minimum Colours it require = 2.
2)
Take a rectangle with out diagonals .
Is a planar graph AND by vertex colouring it requires 2 colors .
THE MINIMUM NO OF COLOURS SUFFICIENT TO This planar graph = 2
ALSO
In any planar graph ,
consider a tree
every tree is a planar grapgh and it is triangle free graph
a tree with n vertices requires 2 colours.
for example
a tree with 100 vertices having 99 edges , it requires minimum 2 (two ) colours.
3)
consider a triangle with 3 vertices & 3 edges
it requires minimum 3 colours
4)
consider a wheel graph of 4 vertices
it requires minimum 4 colours
5)
cosider a complete graph of 4 vertices (k4)
it requires minimum 4 colours
6)cosider a complete graph of 5 vertices (k5)
it requires minimum 5 colours with respect to vertex colour , but k5 is NON PLANAR
IN THE GIVEN QUESTION
ANY plannar graph Means we need to find the maximum no. of vertex-color among all the planner graphs And that maximum no will be the minimum no of color that is sufficient to vertex-color any planer graph.
SO
answer is 4