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 vertice**s (k**_{4})

it requires **minimum 4 colours **

6)cosider a complete graph of 5 vertice**s (k**_{5})

it requires * ***minimum 5 colours with respect to vertex colour , but k**_{5 }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*