Deterministic distributed vertex coloring in polylogarithmic time

Consider an n-vertex graph G = (V,E) of maximum degree ∆, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete rounds. In the distributed vertex coloring problem the objective is to color G with ∆ + 1, or slightly more than ∆ + 1, col-ors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth re-ferred to as running time.) Efficient randomized algorithms for this problem are known for more than twenty years [1, 19]. Specifically, these algorithms produce a (∆+1)-coloring within O(log n) time, with high probability. On the other hand, the best known deterministic algorithm that require