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[Solved]: Hardness of approximation of the 3 colorability problem

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Problem Detail: 

If we have polynomial algorithm that $c$-approximation, $c<\frac{4}{3}$ for graphs that their chromatic number $\geq k$ then $NP=P$, how to prove such statements?

I also have some sort of explanation of this statement: It's NP-hard to separate between graphs that have chromatic number $k$ and chromatic number $c \cdot k$ when $c<\frac{4}{3} \quad \forall k\geq 3$

Asked By : Fayez Abdlrazaq Deab

Answered By : Yuval Filmus

Hint: Consider planar graphs.

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Question Source : http://cs.stackexchange.com/questions/11766

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