The quantum chromatic number, a generalization of the chromatic number, was first defined in relation to the non-local quantum coloring game. We generalize the former by defining the quantum k-distance chromatic number ₊ₐ (G) of a graph G, which can be seen as the quantum chromatic number of the k-th power graph, Gᵏ, and as generalization of the classical k-distance chromatic number ₖ (G) of a graph. It can easily be shown that ₊ₐ (G) ₖ (G). In this paper, we strengthen three classical eigenvalue bounds for the k-distance chromatic number by showing they also hold for the quantum counterpart of this parameter. This shows that several bounds by Elphick et al. J. Combinatorial Theory Ser. A 168, 2019, Electron. J. Comb. 27 (4), 2020 hold in the more general setting of distance-k colorings. As a consequence we obtain several graph classes for which ₊ₐ (G) =₊ (G), thus increasing the number of graphs for which the quantum parameter is known.
Abiad et al. (Tue,) studied this question.