# Faster Deterministic Distributed Coloring Through Recursive List Coloring

## TL;DR

This paper introduces faster deterministic distributed algorithms for graph coloring, significantly improving time complexity for various classes of graphs and coloring variants, using recursive list coloring techniques.

## Contribution

The paper presents novel recursive list coloring algorithms that improve deterministic distributed coloring times for graphs with different properties, including general, bounded arboricity, and bounded neighborhood independence.

## Key findings

- Deterministic $( ext{Delta}+1)$-coloring in $2^{O(
oot{ ext{log} ext{Delta}})} imes ext{log} n$ rounds.
- Efficient $(2+o(1))a$-coloring for graphs with arboricity $a$ in $2^{O(
oot{ ext{log} a})} imes ext{log}^2 n$ rounds.
- Improved $( ext{Delta}+1)$-coloring and $(2 ext{Delta}-1)$-edge coloring algorithms for graphs with bounded neighborhood independence.

## Abstract

We provide novel deterministic distributed vertex coloring algorithms. As our main result, we give a deterministic distributed algorithm to compute a $(\Delta+1)$-coloring of an $n$-node graph with maximum degree $\Delta$ in $2^{O(\sqrt{\log\Delta})}\cdot\log n$ rounds. This improves on the best previously known time complexity for a large range of values of $\Delta$. For graphs with arboricity $a$, we obtain a deterministic distributed algorithm to compute a $(2+o(1))a$-coloring in time $2^{O(\sqrt{\log a})}\cdot\log^2 n$. Further, for graphs with bounded neighborhood independence, we show that a $(\Delta+1)$-coloring can be computed more efficiently in time $2^{O(\sqrt{\log\Delta})} + O(\log^* n)$. This in particular implies that also a $(2\Delta-1)$-edge coloring can be computed deterministically in $2^{O(\sqrt{\log\Delta})} + O(\log^* n)$ rounds, which improves the best known time bound for small values of $\Delta$. All results even hold for the list coloring variants of the problems. As a consequence, we also obtain an improved deterministic $2^{O(\sqrt{\log\Delta})}\cdot\log^3 n$-round algorithm for $\Delta$-coloring non-complete graphs with maximum degree $\Delta\geq 3$. Most of our algorithms only require messages of $O(\log n)$ bits (including the $(\Delta+1)$-vertex coloring algorithms). Our main technical contribution is a recursive deterministic distributed list coloring algorithm to solve list coloring problems with lists of size $\Delta^{1+o(1)}$. Given some list coloring problem and an orientation of the edges, we show how to recursively divide the global color space into smaller subspaces, assign one of the subspaces to each node of the graph, and compute a new edge orientation such that for each node, the list size to out-degree ratio degrades at most by a constant factor on each recursion level.