# Graph Coloring via Degeneracy in Streaming and Other Space-Conscious Models

## TL;DR

This paper presents space-efficient algorithms for graph coloring based on degeneracy across various big data models, nearly matching the theoretical lower bounds and advancing understanding of the complexity involved.

## Contribution

It introduces sublinear algorithms for coloring graphs using degeneracy in multiple models and establishes near-tight lower bounds on their space and query complexity.

## Key findings

- Algorithms color graphs with about κ(G) colors in sublinear time.
- Lower bounds show that achieving similar results requires quadratic space or queries.
- Results are close to optimal given the degeneracy parameter.

## Abstract

We study the problem of coloring a given graph using a small number of colors in several well-established models of computation for big data. These include the data streaming model, the general graph query model, the massively parallel computation (MPC) model, and the CONGESTED-CLIQUE and the LOCAL models of distributed computation. On the one hand, we give algorithms with sublinear complexity, for the appropriate notion of complexity in each of these models. Our algorithms color a graph $G$ using about $\kappa(G)$ colors, where $\kappa(G)$ is the degeneracy of $G$: this parameter is closely related to the arboricity $\alpha(G)$. As a function of $\kappa(G)$ alone, our results are close to best possible, since the optimal number of colors is $\kappa(G)+1$. On the other hand, we establish certain lower bounds indicating that sublinear algorithms probably cannot go much further. In particular, we prove that any randomized coloring algorithm that uses $\kappa(G)+1$ many colors, would require $\Omega(n^2)$ storage in the one pass streaming model, and $\Omega(n^2)$ many queries in the general graph query model, where $n$ is the number of vertices in the graph. These lower bounds hold even when the value of $\kappa(G)$ is known in advance; at the same time, our upper bounds do not require $\kappa(G)$ to be given in advance.