Constant-Time Dynamic Edge Coloring via Dynamic Splitting and LLL

Abstract

We give a dynamic algorithm that maintains a near-optimal edge coloring in constant expected time per update. For every fixed $ε>0$, the algorithm maintains a $(1+ε)Δ$ edge coloring under edge insertions and deletions in $\mathrm{poly}(1/ε)$ worst-case expected update time against an oblivious adversary, provided $Δ\geΔ_ε=\widetildeΘ(1/ε^2)$. The previous constant-time algorithm of Bhattacharya, Costa, Panski, and Solomon (BCPS) required $Δ\ge(\log n/ε^4)^{Θ(\log(1/ε)/ε)}$, where $n$ is the number of vertices. We remove this dependence on $n$, answering their open question. Our approach differs substantially from that of BCPS, which relied on coloring in locally tree-like subgraphs. Instead, we draw on distributed algorithms that organize partial coloring through successive applications of the Lovász Local Lemma (LLL). We develop a dynamic LLL solver with worst-case expected update bounds and a dynamic splitting algorithm that partitions edges into low-degree subgraphs. A key technical contribution is a bottom-up method for iterative splitting, which avoids the logarithmic overhead of maintaining successive levels. It achieves constant expected update time by adaptively choosing which levels to inspect and repair, rather than inspecting every level after each update. We believe these dynamic splitting and LLL tools can be of interest beyond edge coloring.