The quantum chromatic number of a graph 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 , which upper bounds the quantum chromatic number, but is defined under stronger constraints. We study its relation with the chromatic number and the minimum dimension of orthogonal representations . It is known that . We answer three open questions about these relations: we give a necessary and sufficient condition to have , we exhibit a class of graphs such that , and we give a necessary and sufficient condition to have . Our main tools are Kochen–Specker sets, collections of vectors with a traditionally important role in the study of contextuality 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 <etal/> on the quantum chromatic number, we give a family of Kochen–Specker sets of growing dimension.