# chromatic number is np-hard

I'm referring to a question in this book: Algorithms by Jeff Erickson, link:http://jeffe.cs.illinois.edu/teaching/algorithms/notes/J-approx.pdf, in particular, pg.21, Q11.

The author mentioned that finding the chromatic number $$\chi(G)$$ is np-hard, and ask to show that return any integer between $$\chi(G)$$ and $$\chi(G) + 31337$$ is also np-hard. I believe the constant doesn't matter here, my approach is as following: given an instance of finding the chromatic number, we can modify it to finding $$\chi(G)$$ and $$\chi(G) + c$$, return an integer between $$\chi(G)$$ and $$\chi(G) + c$$.

From one suggestion seems like I'm having a wrong approach, any hint that helps me go further is appreciated.

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– D.W.
Dec 6, 2021 at 5:31

It might be easy to find an algorithm that returns any integer between $$\chi(G)$$ and $$\chi(X)+31337$$, and hard to find an algorithm that returns exactly $$\chi(G)$$. (It isn't, but that isn't obvious, and you'll have to prove it.)
• @hhvh, I'm unsure what you mean by "finding an exact range". $\chi(G)$ isn't a range -- it is a single number. I suggest spending some more time working through the details as outlined in my answer.