Quantum chromatic numbers of products of quantum graphs

TL;DR

This paper introduces various product operations for quantum graphs, defines a quantum b-fold chromatic number, and establishes bounds on the quantum chromatic number of these products, advancing the understanding of quantum graph coloring.

Contribution

It defines new product types for quantum graphs and provides bounds on their quantum chromatic numbers, including a novel quantum b-fold chromatic number concept.

Findings

Bounds on quantum chromatic numbers for different quantum graph products

Introduction of quantum b-fold chromatic number for lexicographic products

Framework for analyzing quantum graph coloring in product structures

Abstract

We define the Cartesian, Categorical, and Lexicographic, and Strong products of quantum graphs. We provide bounds on the quantum chromatic number of these products in terms of the quantum chromatic number of the factors. To adequately describe bounds on the lexicographic product of quantum graphs, we provide a notion of a quantum $b$-fold chromatic number for quantum graphs.