Covering of a graph via independent sets 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.
 A: The number that you are considering is the chromatic number of a graph.
A lower bound is the largest size of a clique. 
This number features in some of the most discussed theorems of graph theory,
the perfect graph theorem and the strong perfect graph theorem.
A graph is perfect if every induced subgraph has chromatic number equal maximal clique size.
You will find plenty of material if you search for chromatic number.
A: As has already been pointed out, this is the chromatic number $\chi$. (For example, the assertion that all planar graphs have $ \chi \le 4$ is the famous "Four color theorem.")
You say your graph has maximum degree $d$.  Then in all cases $\chi \le d+1$, and Brooks' theorem gives that in fact $\chi \le d$ as long as $G$ is not a complete graph or an odd cycle. 
A: Since you mentioned that you have a bound on the maximum degree of your graph, you might also want to look at Reed's Conjecture, which states that for any graph $G$,
$\chi(G) \leq \lceil \frac{\Delta (G) + \omega (G) +1}{2}\rceil$.
Here, $\chi$ is the chromatic number, $\Delta$ is the maximum degree, and $\omega$ is the maximum clique number.  
A: If you further specify the class of graphs you're looking at, it may be possible to get a much better bound.  As it is, you cannot really do better than $\Delta+1$ in terms of very easily computable bounds, as mentioned above.
