Hedetniemi-type equality for the quantum chromatic number

Determine whether the quantum chromatic number satisfies the Hedetniemi-type equality χ_q(G × H) = min{χ_q(G), χ_q(H)} for all graphs G and H.

Background

The quantum chromatic number is a homomorphism-monotone graph invariant defined through a graph homomorphism game. Although the Hedetniemi-type equality is known for some special pairs of graphs, its validity for arbitrary graphs remains unresolved.

References

It was proved in that the Hedetniemi-type equality $\chi_q(G \times H) = \min{\chi_q(G), \chi_q(H)}$ holds for some special graphs $G$ and $H$, and it remains an open question whether it holds for all graphs $G$ and $H$.

— A survey on Hedetniemi's conjecture  (2502.16078 - Zhu, 22 Feb 2025) in Section 2.3, paragraph introducing the quantum chromatic number