A recent work by Christiansen, Nowicki, and Rotenberg provides dynamic algorithms for coloring sparse graphs, concretely as a function of the arboricity alpha of the input graph. They give two randomized algorithms: O({alpha} log {alpha}) implicit coloring in poly(log n) worst-case update and query times, and O(min{{alpha} log {alpha}, {alpha} log log log n}) implicit coloring in poly(log n) 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({alpha})-implicit coloring with poly(log n) 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({alpha})-coloring with poly(log n) worst-case update and query times (against an oblivious adversary).

翻译：Christiansen、Nowicki和Rotenberg近期的工作提出了针对稀疏图的动态着色算法，具体以输入图的树宽α作为参数。他们给出了两种随机化算法：在多重对数n的最坏情况更新与查询时间内实现O(α log α)隐式着色，以及在多重对数n的均摊更新与查询时间内实现O(min{α log α, α log log log n})隐式着色（针对非适应性敌手）。我们在颜色数量和时间保证方面改进了这些结果：首先，我们提出一种极其简单的算法，能够在多重对数n的均摊更新与查询时间内计算O(α)隐式着色。其次，作为本研究的主要技术贡献，我们证明时间复杂度的保证可以从均摊强化至最坏情况。具体而言，我们提出了一种动态算法，能够在多重对数n的最坏情况更新与查询时间内实现O(α)隐式着色（针对非适应性敌手）。