Any relation between the size of maximum independent set and the chromatic number on graph of bounded degree? Consider an connected undirected graph $G$ with $n$ vertices and maximum degree $\Delta$. Assume $G$ contains a maximum independent set of size $k$. Is there any relation between the chromatic number $\chi(G)$ and $k$? Say of the form $\chi(G)\le f(\Delta)\frac{n}{k}$ for some function $f$?
Obviously if $f(\Delta)=2\Delta$ then the bound holds.  However, I doubt this is what you meant.  Perhaps you meant $\chi(G) \leq f(\frac n k)$, in which case the answer is no, there is no such function.

Let $\alpha$ be the size of a maximum independent set in $G$.  Then it is easy to see that the fractional chromatic number satisfies the bound $\chi_f(G)\geq \frac n \alpha$.  One such example is the Kneser graph $K_{kb:b}$ for fixed integers $k$ and $b$.  This graph has fractional chromatic number $k = \frac n\alpha$, and chromatic number $(k-2)b+2$.  So for fixed $k$ and large $b$ the answer is no.

(See Section 3.1 of my thesis for more details. http://www.columbia.edu/~ak3074/papers/phdthesis-compact.pdf)