A note on the network coloring game: A randomized distributed $(\Delta +1)$-coloring algorithm

TL;DR

This paper introduces a simple randomized distributed algorithm for graph coloring that guarantees a proper coloring with +1 colors, improving previous strategies that required +2 colors, and operates efficiently in a distributed setting.

Contribution

It presents a modified greedy randomized strategy that achieves proper +1 coloring in a distributed network, reducing the number of colors needed compared to prior methods.

Findings

Achieves proper coloring with +1 colors in a distributed setting.

Operates efficiently with high probability within logarithmic rounds.

Simplifies previous algorithms for network coloring problems.

Abstract

The network coloring game has been proposed in the literature of social sciences as a model for conflict-resolution circumstances. The players of the game are the vertices of a graph with $n$ vertices and maximum degree $\Delta$. The game is played over rounds, and in each round all players simultaneously choose a color from a set of available colors. Players have local information of the graph: they only observe the colors chosen by their neighbors and do not communicate or cooperate with one another. A player is happy when she has chosen a color that is different from the colors chosen by her neighbors, otherwise she is unhappy, and a configuration of colors for which all players are happy is a proper coloring of the graph. It has been shown in the literature that, when the players adopt a particular greedy randomized strategy, the game reaches a proper coloring of the graph within $O(\log(n))$ rounds, with high probability, provided the number of colors available to each player is at least $\Delta+2$. In this note we show that a modification of the aforementioned greedy strategy yields likewise a proper coloring of the graph, provided the number of colors available to each player is at least $\Delta+1$, and results in a simple randomized distributed algorithm for the $(\Delta+1)$-coloring problem..