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.