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Let $G$ be a simple graph with $n$ vertices. Let $\omega(G)$ and $\chi(G)$ denotes the clique number and chromatic number of $G$ respectively. Then

Does $\omega(G)\leq k$ imply $\chi(G)\leq k$ for $n\geq 3$?

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closed as off-topic by Peter Mueller, Ilya Bogdanov, Goldstern, Chris Godsil, Gabriel C. Drummond-Cole Oct 11 '17 at 22:51

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    $\begingroup$ Try a 5-cycle and see for yourself. $\endgroup$ – Gordon Royle Oct 11 '17 at 13:00
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No, even triangle free graphs are counterexample since they can have arbitrary large chromatic number.

https://en.wikipedia.org/wiki/Triangle-free_graph#Coloring_triangle-free_graphs

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No. For the cycle graph $C_5$, $\chi(C_5)=3$, but $\omega(C_5)=2$. So for $k=2$, $\omega(C_5) \le k$ but it is not true that $\chi(C_5) \le k$.

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