Quantum chromatic numbers of some graphs in Hamming schemes

TL;DR

This paper determines the quantum chromatic numbers for a new family of graphs within Hamming schemes, expanding the limited known examples beyond Hadamard graphs and providing bounds for others.

Contribution

It explicitly calculates the quantum chromatic numbers for a class of graphs in Hamming schemes, the second such family with known values, and offers bounds for additional graphs.

Findings

Quantum chromatic numbers are determined for a specific class of Hamming scheme graphs.

Bounds are provided for quantum chromatic numbers of other Hamming scheme graphs.

Results enable calculation of quantum chromatic numbers for product graphs.

Abstract

The study of quantum chromatic numbers of graphs is a hot research topic in recent years. However, the infinite family of graphs with known quantum chromatic numbers are rare, as far as we know, the only known such graphs (except for complete graphs, cycles, bipartite graphs and some trivial cases) are the Hadamard graphs $H_n$ with $2^n$ vertices and $n$ a multiple of $4$. In this paper, we consider the graphs in Hamming schemes, we determined the quantum chromatic numbers of one class of such graphs. Notably, this is the second known family of graphs whose quantum chromatic numbers are explicitly determined except for some cases aforementioned. We also provide some bounds for the quantum chromatic numbers of some other graphs in Hamming schemes. Consequently, we can obtain the quantum chromatic numbers of products of some graphs.