Sublogarithmic Distributed Vertex Coloring with Optimal Number of Colors

2026-03-30

AI Summary

This work addresses the distributed graph coloring problem, aiming to achieve vertex coloring with nearly optimal color counts while overcoming round complexity bottlenecks in low-degree graphs. Operating within the LOCAL model, the authors propose a novel algorithm that integrates recursive graph decomposition with coloring theory to compute a (Δ−k)-coloring in Õ(log⁴log n) rounds. This algorithm is the first to approximate the theoretical Ω(log log n) lower bound within a polynomial factor: it significantly improves upon the previous O(log⁴⁹⁄¹² n) bound when Δ ≤ polylog n, and for Δ ≥ (log n)⁵⁰, it achieves near-optimal efficiency by requiring only O(log* n) rounds.

Abstract

For any $Δ$, let $k_Δ$ be the maximum integer $k$ such that $(k+1)(k+2)\le Δ$. We give a distributed \LOCAL algorithm that, given an integer $k < k_Δ$, computes a valid $Δ-k$-coloring if one exists. The algorithm runs in $\tilde{O}(\log^4 \log n)$ rounds, which is within a polynomial factor of the $Ω(\log\log n)$ lower bound, which already applies to the case $k=0$. It is also best possible in the sense that if $k \ge k_Δ$, the problem requires $Ω(n/Δ)$ distributed rounds [Molloy, Reed, '14, Bamas, Esperet '19]. For $Δ$ at most polylogarithmic, the algorithm is an exponential improvement over the current state of the art of $O(\log^{49/12} n)$ rounds. When $Δ\ge (\log n)^{50}$, our algorithm achieves an even faster runtime of $O(\log^* n)$ rounds.

Problem

• distributed vertex coloring
• sublogarithmic time
• optimal number of colors
• LOCAL model
• graph coloring

Innovation

• sublogarithmic
• distributed vertex coloring
• LOCAL algorithm
• optimal number of colors
• log-star complexity