$(\Delta + 1)$ Vertex Coloring in $O(n)$ Communication

TL;DR

This paper establishes the randomized communication complexity of $(+1)$ vertex coloring in graphs with maximum degree , showing it is (n) bits, which is tight up to constants.

Contribution

The paper provides a randomized protocol for $(+1)$ vertex coloring with optimal communication complexity up to constant factors.

Findings

The protocol uses O(n) bits of communication.

Matching lower bound of (n) bits is established.

The problem's complexity is settled up to constant factors.

Abstract

We study the communication complexity of $(\Delta + 1)$ vertex coloring, where the edges of an $n$-vertex graph of maximum degree $\Delta$ are partitioned between two players. We provide a randomized protocol which uses $O(n)$ bits of communication and ends with both players knowing the coloring. Combining this with a folklore $\Omega(n)$ lower bound, this settles the randomized communication complexity of $(\Delta + 1)$-coloring up to constant factors.