Before revision:  In particular, we obtain an O(\log^* \Delta)-round CONGEST algorithm for ( 2 \Delta -1) -edge coloring when \Delta \ge \log ^2 n, and a poly(\log\log n)-round algorithm  in general.
After revision:  In particular, we obtain an O(\log^* \Delta)-round CONGEST algorithm for ( 2 \Delta -1) -edge coloring when \Delta \ge \log ^{1+1/\log^*n n, and a poly(\log\log n)-round algorithm in general.