Reed's conjecture on some special classes of graphs

Abstract: Reed conjectured that for any graph $G$, $$\chi(G) \leq \lceil \frac{\omega(G)+\Delta(G)+1}{2}\rceil$$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$. In this paper, we verify this conjecture for some special classes of graphs, in particular for subclasses of $P_5$-free graphs or $Chair$-free graphs.