hardness of approximating the chromatic number in graphs with bounded degree
As David pointed out, Khot's paper, "Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring", Theorem 1.6, says it is NP-hard to color $K$-colorable graph with $2^{\Omega((\log K)^2)}$ colors for graphs with degree at most $2^{2^{(\log K)^2}}$, for sufficiently large constant $K$. In other words, for graphs of degree $d$, it is hard to color $2^{\sqrt{\log\log d}}$-colorable graph with $\log d$ colors. 

To get better degree bound, you can probably use ideas from Trevisan's paper "Non-approximability results for optimization problems on bounded degree instances". The key observation is that the graph produced by the FGLSS reduction is a union of complete bipartite subgraphs, and one can replace each of them with a bipartite disperser which is much sparser. Similar idea used in many result such as Chan http://eccc.hpi-web.de/report/2012/110/, Theorem 1.4 / Appendix D.

I think this should give you something like for $2^{\sqrt{c\log d}}$-colorable graphs of degree bounded by $d$, it is NP-hard to color it with $d^{c}$ colors for some constant $0&lt;c&lt;1$.

The degree bound in the paper Michael mentioned is similar to Khot's, namely exponential of the soundness case. Of course the above sparsification approach also improves this, but probably won't give better constant for your purpose.