Dynamic O(Arboricity) Coloring in Polylogarithmic Worst-Case Time

Abstract

A recent work by Christiansen, Nowicki, and Rotenberg [STOC’23] provides dynamic algorithms for coloring sparse graphs, concretely as a function of the graph’s arboricity α. They give two randomized algorithms: O(α logα) implicit coloring in poly(logn) worst-case update and query times, and O(min{α logα, α logloglogn}) implicit coloring in poly(logn) amortized update and query times (against an oblivious adversary). We improve these results in terms of the number of colors and the time guarantee: First, we present an extremely simple algorithm that computes an O(α)-implicit coloring with poly(logn) amortized update and query times. Second, and as the main technical contribution of our work, we show that the time complexity guarantee can be strengthened from amortized to worst-case. That is, we give a dynamic algorithm for implicit O(α)-coloring with poly(logn) worst-case update and query times (against an oblivious adversary).