Distributed Graph Coloring Made Easy

Abstract

In this article, we present a deterministic \(\mathsf {CONGEST}\) algorithm to compute an O ( k Δ)-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 ( k Δ)-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 one-round color reduction algorithms.