Distributed Delta-Coloring under Bandwidth Limitations

We consider the problem of coloring graphs of maximum degree $\mathrm{\Delta }$ with $\mathrm{\Delta }$ colors in the distributed setting with limited bandwidth. Specifically, we give a $\mathsf{p}\mathsf{o}\mathsf{l}\mathsf{y}\log \log n$-round randomized algorithm in the CONGEST model. This is close to the lower bound of $\mathrm{\Omega }(\log \log n)$ rounds from [Brandt et al., STOC '16], which holds also in the more powerful LOCAL model. The core of our algorithm is a reduction to several special instances of the constructive Lovász local lemma (LLL) and the $deg+1$-list coloring problem.