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Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number

The quantum chromatic number of a graph $G$ is sandwiched between its chromatic number and its clique number, which are well known NP-hard quantities. We restrict our attention to the rank-1 quantum chromatic number $χ_q^{(1)}(G)$, which upper bounds the quantum chromatic number, but is defined under stronger constraints. We study its relation with the chromatic number $χ(G)$ and the minimum dimension of orthogonal representations $ξ(G)$. It is known that $ξ(G) \leq χ_q^{(1)}(G) \leq χ(G)$. We answer three open questions about these relations: we give a necessary and sufficient condition to have $ξ(G) = χ_q^{(1)}(G)$, we exhibit a class of graphs such that $ξ(G) < χ_q^{(1)}(G)$, and we give a necessary and sufficient condition to have $χ_q^{(1)}(G) < χ(G)$. Our main tools are Kochen-Specker sets, collections of vectors with a traditionally important role in the study of noncontextuality of physical theories, and more recently in the quantification of quantum zero-error capacities. Finally, as a corollary of our results and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum chromatic number, we give a family of Kochen-Specker sets of growing dimension.