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 ${\chi}'_{4}(2^{r+1}-1)=2^{2r+1}$, ${\chi}'_{5}(2^{r+1})=4^{r+1}$ for any odd number $r\geq3$, 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 ${\chi}'_{d}(n)$.