Distributed Delta-Coloring under Bandwidth Limitations

We consider the problem of coloring graphs of maximum degree Δ with Δ colors in the distributed setting with limited bandwidth. Specifically, we give a polyloglog n-round randomized algorithm in the CONGEST model. This is close to the lower bound of Ω(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.