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+25 votes

The minimum number of colours required to colour the vertices of a cycle with $n$ nodes in such a way that no two adjacent nodes have the same colour is

  1. $2$
  2. $3$
  3. $4$
  4. $n-2 \left \lfloor \frac{n}{2} \right \rfloor+2$
asked in Graph Theory by Veteran (52k points)
retagged by | 2.5k views
plz someone explain clearly.....

2 Answers

+25 votes
Best answer

Chromatic number will be $3$ for when $n$ is odd and will be $2$ when $n$ is even. Option (D) is a representation for this, hence the correct answer

answered by Active (2.4k points)
edited by
I know that Chromatic number will be 3  when n is odd and will be 2 when n is even. But how both of them together give option D. How I will deduce that expression.

@Gupta731 just substitute n = 3 and n = 4 in given equation. You will get the desired answer. 

–1 vote
Dont u think option D is giving answer 2 only!! Bcoz 2(n/2) will give n as ans n-n+2 =2??
answered by (75 points)
$2 \left \lfloor \dfrac n2 \right \rfloor = \begin{cases}n & \text{iff $n$ is even}\\n-1 & \text{iff $n$ is odd}\end{cases}$

So, $n - 2 \left \lfloor \dfrac n2 \right \rfloor = \begin{cases}0 & \text{iff $n$ is even}\\1 & \text{iff $n$ is odd}\end{cases}$

PS: The question has a typo. It says $\left [\dfrac n2 \right ]$ instead of $\left \lfloor \dfrac n2 \right \rfloor$. Let me fix it. :)
Got it..thnx :)

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