New results on two hypercube coloring problems.

Abstract: In this paper, we study the following two hypercube coloring problems: given and , find the minimum number of colors, denoted as (resp. ), needed to color the vertices of the -cube such that any two vertices with Hamming distance at most (resp. exactly ) have different colors. These problems originally arose in the study of the scalability of optical networks. Using methods in coding theory, we show that for any odd number , and give two upper bounds on . The first upper bound improves on that of Kim, Du and Pardalos. The second upper bound improves on the first one for small . Furthermore, we derive an inequality on and . [Copyright &y& Elsevier]