Superfast Coloring in CONGEST via Efficient Color Sampling.

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