Fully Dynamic (Δ +1)-Coloring in O(1) Update Time

The problem of (Δ +1)-vertex coloring a graph of maximum degree Δ has been extremely well studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA’18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized algorithm for maintaining a (Δ +1)-coloring with O(log Δ) expected amortized update time. In this article, we present an improved randomized algorithm for (Δ +1)-coloring that achieves O(1) amortized update time and show that this bound holds not only in expectation but also with high probability. Our starting point is the state-of-the-art randomized algorithm for maintaining a maximal matching (Solomon, FOCS’16). We carefully build on the approach of Solomon, but, due to inherent differences between the maximal matching and (Δ +1)-coloring problems, we need to deviate significantly from it in several crucial and highly nontrivial points.1