For a regular graph with $n$ vertices and maximum degree $\Delta$, it is easy to see that the chromatic number, $\chi\le\frac{n}{2}$ if $\frac{n}{2}\le\Delta\lt n-1$(since a regular graph on $n$ vertices with maximum degree $n-2$ is the complete graph with a one factor removed, which will have each vertex non adjacent to a unique other vertex, which could be given the same color, using the handshaking lemma we get that chromatic number of such a graph is $\frac{n}{2}$)

How could this fact be applied to bound the chromatic number of any non-regular graph with large maximum degree. Does this fact have a well known name, like Reed's theorem, or Brooks' theorem? Thanks beforehand.