## 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.