Improved distributed Delta-coloring

We present a randomized distributed algorithm that computes a $$\Delta $$ -coloring in any non-complete graph with maximum degree $$\Delta \ge 4$$ in $$O(\log \Delta ) + 2^{O(\sqrt{\log \log n})}$$ rounds, as well as a randomized algorithm that computes a $$\Delta $$ -coloring in $$O((\log \log n)^2)$$ rounds when $$\Delta \in [3, O(1)]$$ . Both these algorithms improve on an $$O(\log ^3 n / \log \Delta )$$ -round algorithm of Panconesi and Srinivasan (STOC’93), which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an $$\Omega (\log \log n)$$ round lower bound of Brandt et al. (STOC’16).