Round and Communication Efficient Graph Coloring

TL;DR

This paper introduces communication-efficient randomized and deterministic protocols for graph coloring problems, significantly reducing round complexity and establishing lower bounds in distributed and streaming models.

Contribution

It presents a randomized protocol for vertex coloring with fewer rounds and a deterministic protocol for edge coloring with tight lower bounds, advancing communication complexity in graph coloring.

Findings

Randomized protocol achieves $( ext{max degree}+1)$-vertex coloring with $O(n)$ bits and $O(rac{ ext{log log n} imes ext{log} ext{Delta})$ rounds.

Deterministic protocol computes $(2 ext{Delta}-1)$-edge coloring with $O(n)$ bits and $O(1)$ rounds.

Established an $ ilde{ ext{Omega}}(n)$ lower bound on communication complexity for edge coloring in streaming models.

Abstract

In the context of communication complexity, we explore protocols for graph coloring, focusing on the vertex and edge coloring problems in $n$-vertex graphs $G$ with a maximum degree $\Delta$. We consider a scenario where the edges of $G$ are partitioned between two players. Our first contribution is a randomized protocol that efficiently finds a $(\Delta + 1)$-vertex coloring of $G$, utilizing $O(n)$ bits of communication in expectation and completing in $O(\log \log n \cdot \log \Delta)$ rounds in the worst case. This advancement represents a significant improvement over the work of Flin and Mittal [Distributed Computing 2025], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity. Our second contribution is a deterministic protocol to compute a $(2\Delta - 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication and uses only $O(1)$ rounds. We complement the result with a tight $\Omega(n)$-bit lower bound on the communication complexity of the $(2\Delta-1)$-edge coloring problem, while a similar $\Omega(n)$ lower bound for the $(\Delta+1)$-vertex coloring problem has been established by Flin and Mittal [Distributed Computing 2025]. Our result implies a space lower bound of $\Omega(n)$ bits for $(2\Delta - 1)$-edge coloring in the $W$-streaming model, which is the first non-trivial space lower bound for edge coloring in the $W$-streaming model.