Dynamic $(\Delta + 1)$ Vertex Coloring

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 \S 1 are defined graph coloring, the dynamic model, and the notion of performance of graph algorithms in the dynamic model. In particular $(\Delta + 1)$-coloring, sublinear performance, and oblivious and adaptive adversaries are noted and motivated. In \S 2 the pair of approximately optimal dynamic vertex colorers given in arXiv:1708.09080 are summarized as a warmup for the $(\Delta + 1)$-colorers. In \S 3 the state of the art in dynamic $(\Delta + 1)$-coloring is presented. This section comprises a pair of papers (arXiv:1711.04355 and arXiv:1910.02063) that improve dynamic $(\Delta + 1)$-coloring from the naive algorithm with $O(\Delta)$ expected amortized update time to $O(\log \Delta)$, then to $O(1)$ with high probability. In \S 4 the results in arXiv:2411.04418, which gives a sublinear algorithm for $(\Delta + 1)$-coloring that generalizes oblivious adversaries to adaptive adversaries, are presented.