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We consider the problem of coloring graphs of maximum degree with colors in the distributed setting with limited bandwidth. Specifically, we give a poly n-round randomized algorithm in the CONGEST model. This is close to the lower bound of (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.
Maus et al. (2024) studied this question.