Several recent results from dynamic and sublinear graph coloring are surveyed. This problem is widely studied and has motivating applications like network topology control, constraint satisfaction, and real-time resource scheduling. Graph coloring algorithms are called colorers. In §1 are defined graph coloring, the dynamic model, and the notion of performance of graph algorithms in the dynamic model. In particular $(Δ+ 1)$-coloring, sublinear performance, and oblivious and adaptive adversaries are noted and motivated. In §2 the pair of approximately optimal dynamic vertex colorers given in arXiv:1708.09080 are summarized as a warmup for the $(Δ+ 1)$-colorers. In §3 the state of the art in dynamic $(Δ+ 1)$-coloring is presented. This section comprises a pair of papers (arXiv:1711.04355 and arXiv:1910.02063) that improve dynamic $(Δ+ 1)$-coloring from the naive algorithm with $O(Δ)$ expected amortized update time to $O(\log Δ)$, then to $O(1)$ with high probability. In §4 the results in arXiv:2411.04418, which gives a sublinear algorithm for $(Δ+ 1)$-coloring that generalizes oblivious adversaries to adaptive adversaries, are presented.

翻译：本文综述了动态与亚线性图着色领域近年来的若干研究成果。该问题被广泛研究，在网络拓扑控制、约束满足以及实时资源调度等方面具有重要的应用背景。图着色算法通常被称为着色器。在§1中，我们定义了图着色、动态模型以及动态模型中图算法性能的度量概念。特别地，文中对$(Δ+1)$-着色、亚线性性能、以及非适应性对抗者与适应性对抗者的概念进行了说明与动机阐述。在§2中，我们总结了arXiv:1708.09080中给出的一对近似最优的动态顶点着色器，作为后续$(Δ+1)$-着色器的预备知识。在§3中，我们介绍了动态$(Δ+1)$-着色的最新研究进展。本节涵盖了两篇论文（arXiv:1711.04355与arXiv:1910.02063），它们将动态$(Δ+1)$-着色的性能从朴素算法的$O(Δ)$期望摊还更新时间提升至$O(\log Δ)$，进而以高概率达到$O(1)$。在§4中，我们展示了arXiv:2411.04418中的成果，该工作提出了一种适用于$(Δ+1)$-着色的亚线性算法，并将非适应性对抗者的设定推广至适应性对抗者。