Oddities of quantum colorings

TL;DR

This paper explores the limitations and properties of quantum graph colorings, providing new examples that challenge previous assumptions and reveal differences from classical colorings.

Contribution

It introduces the first known graphs demonstrating the non-comparability of quantum chromatic number and orthogonal rank, and reveals unexpected properties of quantum colorings.

Findings

A graph with a 3D orthogonal representation cannot be quantum 3-colored.

A graph can be quantum 3-colored without a 3D orthogonal representation.

Adding a universal vertex does not necessarily increase the quantum chromatic number.

Abstract

We study quantum analogs of graph colorings and chromatic number. Initially defined via an interactive protocol, quantum colorings can also be viewed as a natural operator relaxation of graph coloring. Since there is no known algorithm for producing nontrivial quantum colorings, the existing examples rely on ad hoc constructions. Almost all of the known constructions of quantum $d$-colorings start from $d$-dimensional orthogonal representations. We show the limitations of this method by exhibiting, for the first time, a graph with a 3-dimensional orthogonal representation which cannot be quantum 3-colored, and a graph that can be quantum 3-colored but has no 3-dimensional orthogonal representation. Together these examples show that the quantum chromatic number and orthogonal rank are not directly comparable as graph parameters. The former graph also provides an example of several interesting, and previously unknown, properties of quantum colorings. The most striking of these is that adding a new vertex adjacent to all other vertices does not necessarily increase the quantum chromatic number of a graph. This is in stark contrast to the chromatic number and many of its variants. This graph also provides the smallest known example (14 vertices) exhibiting a separation between chromatic number and its quantum analog.