We consider the problem of coloring graphs of maximum degree $\Delta$ with $\Delta$ colors in the distributed setting with limited bandwidth. Specifically, we give a $\mathsf{poly}\log\log n$-round randomized algorithm in the CONGEST model. This is close to the lower bound of $\Omega(\log \log n)$ 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 Lov\'asz local lemma (LLL) and the $deg+1$-list coloring problem.

翻译：我们考虑在带宽受限的分布式环境下，对最大度为$\Delta$的图进行$\Delta$种颜色着色的着色问题。具体而言，我们在CONGEST模型中给出了一个$\mathsf{poly}\log\log n$轮的随机化算法。该结果逼近于[Brandt等，STOC '16]中$\Omega(\log \log n)$轮的下界，该下界在更强大的LOCAL模型中也成立。我们算法的核心在于将问题归约到若干构造性Lovász局部引理（LLL）的特殊实例以及$deg+1$-列表着色问题。