Dynamic $(Δ+ 1)$ Vertex Coloring

TL;DR

This work surveys dynamic and sublinear vertex coloring, focusing on maintaining a $(Δ+1)$-coloring under graph updates. It charts a progression from trivial $O(Δ)$ update time to $O(\log Δ)$ and then to $O(1)$ with high probability in the oblivious-adversary setting, via a level-based recoloring framework. It also presents sublinear results for adaptive adversaries, leveraging phase-based sparse-dense decompositions and randomized recoloring, achieving sublinear update times such as tilde{O}(n^{8/9}) in this more challenging model. Collectively, the results advance the practical feasibility of dynamic coloring for real-time network control and constraint-satisfaction tasks, with clear algorithmic techniques for balancing recolorings, palette sizes, and robustness to adversarial inputs.

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 §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.

Figures (5)

• Figure 1: $\ell(v)$ for each $v \in V$, with up-neighbors $\mathcal{H}(v)$, same-level neighbors $\mathcal{S}(v) \subseteq \mathcal{H}(v)$, and down-neighbors $\mathcal{L}(v)$.
• Figure 2: Classification of levels based on epoch types. The small colored nodes represent individual epochs, colored by their termination cause. Vertex $v$ is shown at level $\ell(v)$, with weak dashed lines indicating its neighborhood in different levels.
• Figure 3: Epochs $\mathcal{E}_1, \mathcal{E}_2, \mathcal{E}_3$ for vertex $x = v(\mathcal{E}_1) = v(\mathcal{E}_2) = v(\mathcal{E}_3)$. $\mathcal{E}$ corresponds to the maximal time interval between consecutive recolorings of $x$. The cost of an epoch with level $\ell(\mathcal{E}) = -1$ ($\mathcal{E}_2$) is charged to the preceding epoch ($\mathcal{E}_1$). Here $\mathcal{E}_3$ could be induced.
• Figure 4: Example pseudo-duration, first in the oblivious case as in $\mathcal{A}_{\log}$, then how an adaptive adversary would exploit pseudo-duration to try to decrease epoch length.
• Figure 5: HSS decomposition scheme. $V(G)$ is partitioned into sparse vertices $V_S$ and dense vertices $V_D$. $V_D$ is a disjoint union of almost-cliques $C_1, C_2, \dots, C_k$. Vertex $v \in C_1$ illustrates the property of being connected to almost all of its own clique with few edges leaving it.

Theorems & Definitions (2)

• Lemma 3.3
• proof