A coloring problem on the n-cube

In this paper, we consider a coloring problem on the n-cube that arises in the study of scalability of optical networks. Let χk̄(n) be the minimum number of colors needed to color the vertices of the n-cube so that every two vertices with Hamming distance ⩽k have different colors. We show that for k=3, 2n⩽χ3̄(n)⩽2⌈log2 n⌉+1. We also provide upper and lower bounds on χk̄(n) for general k.