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+15 votes
1.3k views

The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same color, is

  1. 2
  2. 3
  3. 4
  4. 5
asked in Graph Theory by Veteran (69k points) | 1.3k views

graph is planar so we can directly say it requires not more than 4 color

this graph has also odd length cycle for that 3 color requires

also this graph has even length cycle for that 2 color 

but i have a doubt here :

if graph is very complex and if both the even and odd length cycle appears as well as planar solution then whts the correct aproach to find color requirement in graph 

same situation with this https://gateoverflow.in/3263/gate2008-it-3

@Bikram sir which approach ?pls help

 

The chromatic number of a planar graph is no more than four.

We can use this approach here.

as it is maximum 4 , then 4 colors require for complex kind of graphs where both the even and odd length cycle appears as well as planar solution ..

5 Answers

+20 votes
Best answer

4 colors are required to color the graph in the prescribed way.

answer = option C

answered by Veteran (31k points)
selected by
Practical approach :)
Here we also need to prove that the given graph can't be colored with less than 4 colors.

As it consists cycle with odd vertices it requires at least 3 colors.

But if you try to color them with 3 colors it is not possible to color them with 3 colors.

More ever any planar graph can be colored with 4 colors (Four color theorem).

This might help ...

 

+6 votes

Ans C 

answered by Veteran (10.9k points)
+2 votes
note:-The max degree of the vertex is 4 so we need atmost 4 colours to colour the graph

answer is C) 4 only
answered by Boss (5.9k points)
edited by

This is not a correct theorem/result.Counter ex: Degree of every vertex in Graph K5 is 4 (max degree is 4) but we need 5 colors to color the graph.

correct theorem: If every vertex in G has degree at most d then G admits a vertex coloring using d+1 colors. 

–1 vote
4 colors
answered by Boss (7.8k points)
–5 votes
The max degree of the vertex is 4 so we need atleast 5 colours to colour the graph
answered by Veteran (14.3k points)
4 colours are enough here !
It is a planar graph so we need only 4 colour.

 Theorem: If every vertex in G has degree at most d then G admits a vertex coloring using d+1 colors. 

But here d+1 is not the min value.



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