Dynamic Algorithms for Graph Coloring

TL;DR

This paper introduces new dynamic algorithms for graph coloring that efficiently maintain proper vertex and edge colorings during graph updates, improving on previous methods in terms of speed and determinism.

Contribution

It presents three novel algorithms: a randomized $( ext{Delta}+1)$-vertex coloring with $O( ext{log} ext{Delta})$ expected update time, a deterministic $(1+o(1)) ext{Delta}$-vertex coloring with polylogarithmic update time, and a simple deterministic $(2 ext{Delta}-1)$-edge coloring with $O( ext{log} ext{Delta})$ worst-case update time.

Findings

Randomized algorithm maintains $( ext{Delta}+1)$-vertex coloring efficiently.

Deterministic algorithm maintains near-optimal vertex coloring with polylogarithmic update time.

Deterministic edge coloring algorithm improves worst-case update time over previous methods.

Abstract

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for $(\Delta+1)$- vertex coloring and $(2\Delta-1)$-edge coloring in a graph with maximum degree $\Delta$. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a $(\Delta+1)$-vertex coloring with $O(\log \Delta)$ expected amortized update time. (2) We present a deterministic algorithm which maintains a $(1+o(1))\Delta$-vertex coloring with $O(\text{poly} \log \Delta)$ amortized update time. (3) We present a simple, deterministic algorithm which maintains a $(2\Delta-1)$-edge coloring with $O(\log \Delta)$ worst-case update time. This improves the recent $O(\Delta)$-edge coloring algorithm with $\tilde{O}(\sqrt{\Delta})$ worst-case update time by Barenboim and Maimon.