Distributed Graph Coloring Made Easy

Abstract

In this paper, we present a deterministic \(\mathsf {CONGEST} \) algorithm to compute an O(kΔ)-vertex coloring in O(Δ/k) + log *n rounds, where Δ is the maximum degree of the network graph and k ≥ 1 can be freely chosen. The algorithm is extremely simple: Each node locally computes a sequence of colors and then it tries colors from the sequence in batches of size k. Our algorithm subsumes many important results in the history of distributed graph coloring as special cases, including Linial’s color reduction [Linial, FOCS’87], the celebrated locally iterative algorithm from [Barenboim, Elkin, Goldenberg, PODC’18], and various algorithms to compute defective and arbdefective colorings. Our algorithm can smoothly scale between several of these previous results and also simplifies the state of the art (Δ + 1)-coloring algorithm. At the cost of losing some of the algorithm’s simplicity we also provide a O(kΔ)-coloring algorithm in \(O(\sqrt {\Delta /k})+\log ^{*} n \) rounds. We also provide improved deterministic algorithms for ruling sets, and, additionally, we provide a tight characterization for 1-round color reduction algorithms.