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
χq(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)≤χq(1)​(G)≤χ(G). We
answer three open questions about these relations: we give a necessary and
sufficient condition to have ξ(G)=χq(1)​(G), we exhibit a class of
graphs such that ξ(G)<χq(1)​(G), and we give a necessary and
sufficient condition to have χq(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.Comment: 12 page