Relaxed two-coloring of cubic graphs

We show that any graph of maximum degree at most 3 has a two-coloring such that one color-class is an independent set, while the other color-class induces monochromatic components of order at most 750. On the other hand, for any constant C, we exhibit a 4-regular graph such that the deletion of any independent set leaves at least one component of order greater than C. Similar results are obtained for coloring graphs of given maximum degree with k+ℓ colors such that k parts form an independent set and ℓ parts span components of order bounded by a constant. A lot of interesting questions remain open.