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Reed conjectured that for every graph, $\chi \leq \left \lceil \frac{\Delta +\omega + 1}{2} \right \rceil$ holds, where $\chi$, $\omega$ and $\Delta$ denotethe chromatic number, clique number and maximum degree of the graph,respectively. We develop an algorithm which takes a hypothetical counterexampleas input. The output discloses some hidden structures closely related to highvertex degrees. Consequently, we deduce two graph classes where Reed'sConjecture holds: One contains all graphs in which the vertices of degree atleast $5$ form a stable set. The other contains all graphs in which everyinduced cycle of odd length contains a vertex of at most degree 3.

A note on Reed's Conjecture about $\omega$, $\Delta$ and $\chi$ with respect to vertices of high degree