Fast Distributed Brooks' Theorem.

We give a randomized ∆-coloring algorithm in the LOCAL model that runs in poly log log n rounds, where n is the number of nodes of the input graph and ∆ is its maximum degree. This means that randomized ∆-coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, poly log log n, given the known Ω(log _∆ log n) lower bound [Brandt et al., STOC'16]. Our main technical contribution is a constant time reduction to a constant number of (deg + 1)-list coloring instances, for ∆ = ω(log ⁴ n), resulting in a poly log log n-round CONGEST algorithm for such graphs. This reduction is of independent interest for other settings, including providing a new proof of Brooks' theorem for high degree graphs, and leading to a constant-round Congested Clique algorithm in such graphs. When ∆ = Ω(log ²² n), our algorithm even runs in O(log ^∗ n) rounds, showing that the base in the Ω(log _∆ log n) lower bound is unavoidable. Previously, the best LOCAL algorithm for all considered settings used a logarithmic number of rounds. Our result is the first CONGEST algorithm for ∆-coloring non-constant degree graphs.