Distributed Recoloring

TL;DR

This paper studies the problem of distributed graph recoloring, focusing on how many communication rounds and steps are needed to recolor networks while maintaining proper coloring, especially with one extra color in specific graph classes.

Contribution

It introduces the concept of distributed recoloring and analyzes the complexity of recoloring schedules with one extra color in trees, 3-regular graphs, and toroidal grids.

Findings

Recoloring with one extra color is feasible in certain graph classes.

The number of communication rounds depends on the graph class and coloring constraints.

Efficient recoloring schedules can be designed for trees, 3-regular graphs, and toroidal grids.

Abstract

Given two colorings of a graph, we consider the following problem: can we recolor the graph from one coloring to the other through a series of elementary changes, such that the graph is properly colored after each step? We introduce the notion of distributed recoloring: The input graph represents a network of computers that needs to be recolored. Initially, each node is aware of its own input color and target color. The nodes can exchange messages with each other, and eventually each node has to stop and output its own recoloring schedule, indicating when and how the node changes its color. The recoloring schedules have to be globally consistent so that the graph remains properly colored at each point, and we require that adjacent nodes do not change their colors simultaneously. We are interested in the following questions: How many communication rounds are needed (in the LOCAL model of distributed computing) to find a recoloring schedule? What is the length of the recoloring schedule? And how does the picture change if we can use extra colors to make recoloring easier? The main contributions of this work are related to distributed recoloring with one extra color in the following graph classes: trees, $3$-regular graphs, and toroidal grids.