The packing chromatic number of hypercubes

The packing chromatic number χρ(G) of a graph G is the smallest integer k needed to proper color the vertices of G in such a way that the distance in G between any two vertices having color i be at least i + 1. Goddard et al. [9] found an upper bound for the packing chromatic number of hypercubes Qn. Moreover, they compute χρ(Qn) for n ≤ 5 leaving as an open problem the remaining cases. In this paper, we obtain a better upper bound for χρ(Qn) and we improve the lower bounds for χρ(Qn) for 6 ≤ n ≤ 11. In particular we compute the exact value of χρ(Qn) for 6 ≤ n ≤ 8.