GATE IT 2008 | Question: 3

Question

What is the chromatic number of the following graph?

 

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Comments

some actual colouring :p

Answer 1

 

The chromatic number of a graph  is the smallest number of colors needed to color the vertices so that no two adjacent vertices share the same color.

Hence minimum number of colors needed to color given graph is equal to $3$

For odd length cycles we need minimum $3$ colors for vertex coloring and for even length cycles we need just $2$. 

Answer is $B$

Comments

how do we differentiate between regular and planar graph to calculate chromatic number.

Like for planar graph we have chromatic number 4. whereas if we have odd cycles chromatic number = 3 else for even it is 2.

Please help.

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chromatic number of cycle graph if it is planar

3 if n is odd

2 if n is even

chromatic number of wheel graph if it is planar

3 if n is odd

4 if n is even

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The Four Color theorem

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

Using this we can rule out some option directly. 

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For odd length cycles we need minimum 3 colors for vertex coloring and for even length cycles we need just 2. 

How is this information used to solve the given question ? Can someone elaborate on it ? 

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In the given graph , there is a cycle of length 5, 5 is odd, so, at-least 3 colors are needed. Now try to color the vertexes using 3  colors such that no adjacent vertex gets same color(you will be able it using 3 colors)

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@Hemant Parihar It’s not a theorem it’s a conjecture.

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@raja11sep Appel and Haken successfully delivered the proof of the Four Color Theorem in 1976, eliminating it as a conjecture being posed in the 1850s.

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I think it is the first proof which was proved by computer .

https://en.wikipedia.org/wiki/Computer-assisted_proof#:~:text=In%201976%2C%20the%20four%20color,verified%20using%20a%20computer%20program.

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thanks

Answer 2

$chromatic\ number(\chi) \ge \frac{total\ number\ of\ vertices(n)}{Independent\ set(\alpha)}$

$n=9\ and\ \alpha=4$

$\chi = \left \lceil \frac{9}{4} \right \rceil = 3$

$and\ also$

 

$so\ it\ cannot\ be\ 4\ and\ answer\ is\ (b)$

Comments

What is the independent set here?

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R_Ayush777

yellow , green and red vertices are three independent sets here!

Answer 3

So 3 color used hence chromatic number =3 option B

Comments

answer is $3$ option B)

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Edited Thanks  @reena_kandari

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Is there any problem with answer @puja mishra @rajoramanoj

Answer 4

hence, 3 colors required. so chromatic no is 3.