Let $G$ be a simple graph such that $\omega(G)\leq\lfloor\frac{\Delta(G)+1}{2}\rfloor+1$ where $\Delta(G)$ is the maximal degree of $G$. Is it true that
\begin{equation}
\chi(G)\leq \lfloor\frac{\Delta(G)+1}{2}\rfloor+2?
\end{equation}
A similar problem can be found [here](http://onlinelibrary.wiley.com/doi/10.1002/(SICI)1097-0118(199804)27:4%3C177::AID-JGT1%3E3.0.CO;2-K/full), where, ignoring the exceptional case of odd cycles, it is conjectured that



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> **[Conjecture (Reed)](http://www.openproblemgarden.org/op/reeds_omega_delta_and_chi_conjecture).** For any graph $G$ of maximum degree $\Delta(G)$, $\chi(G)$ is at most $\lceil\frac{\Delta(G)+1+\omega(G)}{2}\rceil$.
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**UPDATE:** In the case of graphs that satisfy $\omega(G)\leq\lfloor\frac{\Delta(G)+1}{2}\rfloor+1$, the statement in my problem is stronger than Reed's conjecture. For example suppose $\Delta=100$ and $\omega=50$; then by Reed's conjecture we need $76$ colors, while if my conjecture is true, we need just $52$ colors!



Upper bound for chromatic number of graphs with $\omega(G)\leq\lfloor\frac{\Delta(G)+1}{2}\rfloor+1$
As far as I know, your conjecture is an open problem, because Reed's conjecture is still open for $\omega=2$. For $\omega=2$, your conjecture is essentially equivalent to Reed's conjecture. That is, up to rounding, both conjectures assert that every triangle-free graph has chromatic number at most $\Delta/2+2$. 


As mentioned by Fedor Petrov, a bound of $9\Delta / 
\log \Delta$ was proved by Johansson (even for the list version) in this case. 


One can try to lower the constant term of $9$, by restricting to large $\Delta$. The state of the art in this direction is a recent [paper](https://arxiv.org/abs/1701.09133) of Mike Molloy who proves that for every $\epsilon >0$, there exists $\Delta\_\epsilon$ such that every graph triangle-free $G$ with maximum degree at least $\Delta\_\epsilon$ has list chromatic number at most $(1+\epsilon) \frac{\Delta(G)}{\log \Delta(G)}$.


On the other hand, if we only want a linear bound in $\Delta$, but with the constant term optimized, the best result (as far as I know) is by Kostochka. He proved that triangle-free graphs have chromatic number at most $2\Delta/3+2$. I could not track down a reference, but this is well-known. See for example, Section 2.3 of [Andrew King's](http://andrewdouglasking.com/) PhD thesis. 


In summary, your conjecture is not known to be true even for $\omega=2$.


