Before revision:  It is known that for every real \alpha>1 and integer \Delta, a fractional coloring of total weight at most \alpha(\Delta+1) can be obtained deterministically in a single round in graphs of maximum degree \Delta, in the LOCAL model of computation.
After revision:  It was proved by Hasemann, Hirvonen, Rybicki and Suomela (2016) that for every real \alpha>1 and integer \Delta, a fractional coloring of total weight at most \alpha(\Delta+1) can be obtained deterministically in a single round in graphs of maximum degree \Delta, in the LOCAL model of computation.