We present a procedure for efficiently sampling colors in the CONGEST {\congest} model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to \Theta(\log n) semi-random colors unused by their neighbors in O(1) rounds, even in the distance-2 setting. This yields algorithms with O(\log^* \Delta) complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. 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. The sampling procedure is inspired by a seminal result of Newman in communication complexity.
We present a procedure for efficiently sampling colors in the {\congest} model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to \Theta(\log n) semi-random colors unused by their neighbors in O(1) rounds, even in the distance-2 setting. This yields algorithms with O(\log^* \Delta) complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an O(\log^* \Delta)-round CONGEST algorithm for ( 1+\epsilon) \Delta -edge coloring when \Delta \ge \log ^{1+1/\log^*n n, and a poly(\log\log n)-round algorithm for (2\Delta-1)-edge coloring in general. The sampling procedure is inspired by a seminal result of Newman in communication complexity.
D
CONGEST
R
2
1+\epsilon)
D
-1)
R
^2
^{1+1/\log^*n
A
for (2\Delta-1)-edge coloring