On the packing chromatic number of hypercubes

The packing chromatic number χ ρ ( 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. [8] found an upper bound for the packing chromatic number of hypercubes Q n . Moreover, they compute χ ρ ( Q n ) for n ⩽ 5 leaving as an open problem the remaining cases. In this paper, we obtain a better upper bound for χ ρ ( Q n ) and we compute the exact value of χ ρ ( Q n ) for 6 ⩽ n ⩽ 8 .