Before revision:  Moreover, we improve on classical bounds on the chromatic number by considering the fractional chromatic number instead, without significantly increasing the output size and the round complexity of the existing algorithms, while finding a fractional coloring of total weight at most \Delta in this case requires \Omega(\log \log n) rounds for randomized algorithms and \Omega( \log n) rounds for deterministic algorithms.
After revision:  More precisely, we show that for any fixed \epsilon > 0 and \Delta, a fractional coloring of total weight at most \Delta+\epsilon can be found in O(\log^*n) rounds in graphs of maximum degree \Delta with no K_{\Delta+1, while finding a fractional coloring of total weight at most \Delta in this case requires \Omega(\log \log n) rounds for randomized algorithms and \Omega( \log n) rounds for deterministic algorithms.