  In \cite{reed97}, Reed conjectures that the inequality $\chi (G) \leq \left
\lceil \textstyle {1/2} (\omega (G) + \Delta (G) + 1) \right \rceil$ holds for
any graph $G$. We prove this holds for a graph $G$ if $\bar{G}$ is
disconnected. From this it follows that the conjecture holds for graphs with
$\chi(G) > \left \lceil \frac{|G|}{2} \right \rceil$. In addition, the
conjecture holds for graphs with $\Delta(G) \geq |G| - \sqrt{|G| + 2\alpha(G) +
1}$. In particular, Reed's conjecture holds for graphs with $\Delta(G) \geq |G|
- \sqrt{|G| + 7}$. Using these results, we proceed to show that if $|G|$ is an
even order counterexample to Reed's conjecture, then $\bar{G}$ has a 1-factor.
Hence, for any even order graph $G$, if $\chi(G) > \textstyle {1/2}(\omega(G) +
\Delta(G) + 1) + 1$, then $\bar{G}$ is matching covered.
