Consider any regular graph $G$ with order $n$ and size $E$ and maximum degree $\Delta$. Now, we give a $\Delta+1$ coloring to the vertices such that each vertex and its neighbors receive distinct colors.

Consider a color class of independent vertices in such a coloring. If we remove the color class, then do we have a regular subgraph with with maximum degree $\Delta-1$?

I think yes. It is easily seen to be true in the case of complete graphs. It can also be extended, I think, to graphs with $\Delta\ge\frac{n}{2}$, since each color class in such a coloring would have at most $2$ vertices. But, given any regular graph, is the claim true? Thanks beforehand.