Quantum Chromatic Number of Subgraphs of Orthogonality Graphs and the Distance-2 Hamming Graph

TL;DR

This paper advances the understanding of quantum chromatic numbers by calculating them for specific subgraphs of orthogonality graphs and the distance-2 Hamming graph, employing combinatorial design techniques.

Contribution

It extends known results by determining quantum chromatic numbers for new classes of subgraphs and the distance-2 Hamming graph using combinatorial methods.

Findings

Quantum chromatic number of certain orthogonality subgraphs determined.

Quantum chromatic number of distance-2 Hamming graphs established for infinitely many lengths.

Abstract

The determination of the quantum chromatic number of graphs has attracted considerable attention recently. However, there are few families of graphs whose quantum chromatic numbers are determined. A notable exception is the family of orthogonality graphs, whose quantum chromatic numbers are fully determined. In this paper, we extend these results by determining the exact quantum chromatic number of several subgraphs of the orthogonality graphs. Using the technique of combinatorial designs, we also determine the quantum chromatic number of the distance-2 Hamming graph, whose edges consist of binary vectors of Hamming distance 2, for infinitely many length.