Eigenvalue bounds for the quantum chromatic number of graph powers

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 χkq(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 χₖ(G) of a graph. It can easily be shown that χkq(G) ≤ χₖ(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 χkq(G)=χₖ(G), thus increasing the number of graphs for which the quantum parameter is known.