LinearTime and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces
Abstract
In 1994, Thomassen proved that every planar graph is 5listcolorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3listcolorable. His proofs naturally lead to quadratictime algorithms to find such colorings. Here, we provide the first such lineartime algorithms to find such colorings. For a fixed surface S, Thomassen showed in 1997 that there exists a lineartime algorithm to decide if a graph embedded in S is 5colorable and similarly in 2003 if a graph of girth at least five embedded in S is 3colorable. Using the theory of hyperbolic families, the author and Thomas showed such algorithms exist for listcolorings. Dvorak and Kawarabayashi actually gave an $O(n^{O(g+1)})$time algorithm to find such colorings (if they exist) in nvertex graphs where g is the Euler genus of the surface. Here we provide the first such algorithm whose exponent does not depend on the genus; indeed, we provide a lineartime algorithm. In 1988, Goldberg, Plotkin and Shannon provided a deterministic distributed algorithm for 7coloring nvertex planar graphs in $O(\log n)$ rounds. In 2018, Aboulker, Bonamy, Bousquet, and Esperet provided a deterministic distributed algorithm for 6coloring nvertex planar graphs in $O(\log^3 n)$ rounds. Their algorithm in fact works for 6listcoloring. They also provided an $O(\log^3 n)$round algorithm for 4listcoloring trianglefree planar graphs. Chechik and Mukhtar independently obtained such algorithms for ordinary coloring in $O(\log n)$ rounds, which is best possible in terms of running time. Here we provide the first polylogarithmic deterministic distributed algorithms for 5coloring nvertex planar graphs and similarly for 3coloring planar graphs of girth at least five. Indeed, these algorithms run in $O(\log n)$ rounds, work also for listcolorings, and even work on a fixed surface (assuming such a coloring exists).
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.03723
 Bibcode:
 2019arXiv190403723P
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 Computer Science  Data Structures and Algorithms;
 05C15
 EPrint:
 20 pages, revised version