I suspect that a topic such as this may have been considered before: if so, I hope that someone can point me to a reference on the subject.

I have a graph *G* with an upper bound *d* on its maximum degree. Define an *independent-set cover* for *G* to be a family of independent sets in *G*, whose union is *V(G)*. Is there a non-trivial upper bound for the smallest independent-set cover of *G*? [That is: I would like the number of independent sets in the family to be as small as possible; the size of any particular independent set within that family is unimportant.]

I'm also interested in the case where the degrees of the vertices may differ substantially from the maximum degree.

**Edited to add:** thanks to those who pointed out that the size of the smallest independent-set cover is just the chromatic number (by definition). I'm not sure how I managed to avoid noticing that this is what I was asking about, except that I probably think of colourings too much in terms of vertex-labellings and not often enough in terms of vertex partitions.