New Results on Two Hypercube Coloring Problems

TL;DR

This paper investigates hypercube vertex coloring problems related to Hamming distances, providing exact values for certain cases, improved bounds, and inequalities, with applications to optical network scalability.

Contribution

It introduces new exact results, improved upper bounds, and inequalities for hypercube coloring numbers using coding theory methods.

Findings

Exact values of ${}'_{4}(2^{r+1}-1)$ and ${}'_{5}(2^{r+1})$

Two improved upper bounds on ${}_{d}(n)$

Derived inequality relating ${}_{d}(n)$ and ${}'_{d}(n)

Abstract

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 ${\chi}'_{d}(n)$.