A regular independence induced graph in a $\Delta+1$ coloring

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.