The exact quantum chromatic number of Hadamard graphs

TL;DR

This paper determines the exact quantum chromatic number of Hadamard graphs of order 2^N for N divisible by 4, using advanced bounds and conjugacy class graph techniques, including products of such graphs.

Contribution

It introduces a novel approach combining upper and lower bounds with conjugacy class graph results to compute exact quantum chromatic numbers for Hadamard graphs and their products.

Findings

Exact quantum chromatic number for Hadamard graphs of order 2^N (N multiple of 4)

Bounds on quantum chromatic numbers of Hadamard graph products

Application of conjugacy class graph techniques to quantum graph coloring

Abstract

We compute the exact value of the quantum chromatic numbers of Hadamard graphs of order $n=2^N$ for $N$ a multiple of $4$ using the upper bound derived by Avis, Hasegawa, Kikuchi, and Sasaki, as well as an application of the Hoffman-like lower bound of Elphick and Wocjan that was generalized by Ganesan for quantum graphs. As opposed to prior computations for the lower bound, our approach uses Ito's results on conjugacy class graphs allowing us to also find bounds on the quantum chromatic numbers of products of Hadamard graphs. In particular, we also compute the exact quantum chromatic number of the categorical product of Hadamard graphs.