# An upper bound for graph chromatic number related with $ω^2$ [closed]

In graph coloring I have heared the inequality $\chi(G) \le \omega(G) ^2$ where $\chi$ is graph chromatic number and $\omega$ is graph clique number in an arbitary graph G but actually I'm not sure whether it's true as far as I wasn't succesful proving it.
I would appreciate if you could clarify.

## closed as unclear what you're asking by Shaun, user91500, Claude Leibovici, Shailesh, Arnaud D.Apr 10 '17 at 15:08

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• What do you mean by the symbol $\omega^2$? – Laars Helenius Apr 9 '17 at 19:04
• @LaarsHelenius In graph theory $\omega$ is the maximal sub graph which is a clique. – Smile Apr 9 '17 at 19:08

It's false: the Mycielski graph construction (Wikipedia link) shows that there exist graphs $G$ with $\chi(G)$ arbitrarily large but $\omega(G)=2$.
We do have $\chi(G) = \omega(G)$ when $G$ is a perfect graph. (This is sort of a by-definition claim, but perfect graphs are now known to be characterized by the strong perfect graph theorem, giving my statement some content.)
And, of course, we always have $\chi(G) \ge \omega(G)$.