Dynamic $(\Delta + 1)$ Vertex Coloring

TL;DR

This paper surveys recent advances in dynamic graph coloring, focusing on algorithms for efficiently maintaining a proper $( + 1)$ vertex coloring under various adversarial models, with significant improvements in update times.

Contribution

It summarizes state-of-the-art algorithms for dynamic $( + 1)$-coloring, highlighting recent progress from naive to near-constant update times and extending algorithms to adaptive adversaries.

Findings

Expected amortized update time reduced to O(log Δ)

High probability update time achieved as O(1)

Sublinear algorithms applicable to adaptive adversaries

Abstract

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.