Distributed coloring in sparse graphs with fewer colors

TL;DR

This paper presents a new distributed algorithm for coloring sparse graphs, especially planar graphs, with fewer colors and in fewer rounds than previous methods, advancing the efficiency of distributed graph coloring.

Contribution

It introduces a distributed algorithm that colors planar graphs with 6 colors in polylogarithmic rounds, improving upon prior algorithms that used more colors or had higher complexity.

Findings

Coloring planar graphs with 6 colors in polylogarithmic rounds.

No distributed algorithm can color all planar graphs with 4 colors in o(n) rounds.

The algorithm extends to list-coloring and general sparse graphs.

Abstract

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in $\mbox{polylog}(n)$ rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.