Optimal Distributed Coloring Algorithms for Planar Graphs in the LOCAL model

TL;DR

This paper introduces an optimal distributed algorithm for 6-coloring planar graphs in the LOCAL model and establishes a near-linear lower bound for 4-coloring, resolving key open problems in distributed graph coloring.

Contribution

The paper presents the first near-optimal distributed algorithm for 6-coloring planar graphs and proves a linear lower bound for 4-coloring, fully resolving these problems in the LOCAL model.

Findings

Optimal $O( ext{log} n)$ time algorithm for 6-coloring

Linear lower bound for 4-coloring planar graphs

Novel technique for removing small structures in coloring algorithms

Abstract

In this paper, we consider distributed coloring for planar graphs with a small number of colors. We present an optimal (up to a constant factor) $O(\log{n})$ time algorithm for 6-coloring planar graphs. Our algorithm is based on a novel technique that in a nutshell detects small structures that can be easily colored given a proper coloring of the rest of the vertices and removes them from the graph until the graph contains a small enough number of edges. We believe this technique might be of independent interest. In addition, we present a lower bound for 4-coloring planar graphs that essentially shows that any algorithm (deterministic or randomized) for $4$-coloring planar graphs requires $\Omega(n)$ rounds. We therefore completely resolve the problems of 4-coloring and 6-coloring for planar graphs in the LOCAL model.