Improved Distributed $\Delta$-Coloring

We present a randomized distributed algorithm that computes a $\Delta$-coloring in any non-complete graph with maximum degree $\Delta \geq 4$ 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'1993], 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].