New results on two hypercube coloring problems

In this paper, we study the following two hypercube coloring problems: given n and d, find the minimum number of colors, denoted as chi(d)'(n) (resp. chi(d)(n)), needed to color the vertices of the n-cube such that any two vertices with Hamming distance at most d (resp. exactly d) have different colors. These problems originally arose in the study of the scalability of optical networks. Using methods in coding theory, we show that chi(5)'(2(r+1)) = 4(r+1) for any odd number r >= 3, and give two upper bounds on chi(d)(n). The first upper bound improves on that of Kim, Du and Pardalos. The second upper bound improves on the first one for small n. Furthermore, we derive an inequality on chi(d)(n) and chi(d)'(n). (C) 2013 Elsevier B.V. All rights reserved.