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(logn) 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 Ω(n) rounds. We therefore completely resolve the problems of 4-coloring and 6-coloring for planar graphs in the LOCAL model.
@article{arxiv.1804.00137,
title = {Optimal Distributed Coloring Algorithms for Planar Graphs in the LOCAL model},
author = {Shiri Chechik and Doron Mukhtar},
journal= {arXiv preprint arXiv:1804.00137},
year = {2018}
}