The square of the 9-hypercube is 14-colorable

TL;DR

This paper investigates the coloring of the 9-dimensional hypercube to prevent vertices within a certain distance from sharing the same color, providing improved bounds through computational methods.

Contribution

It improves the upper bound on the minimum number of colors needed to color the 9-hypercube with distance constraints, using computer search techniques.

Findings

Established that 13 ≤ χ̄₂(9) ≤ 14

Improved previous upper bounds for hypercube coloring

Demonstrated computational approach for coloring bounds

Abstract

The $n$-hypercube, denoted by $Q_n$, has a vertex for each bit string of length $n$ with two vertices adjacent whenever their Hamming distance is one. The minimum number of colors needed to color $Q_n$ such that no two vertices at a distance at most $k$ receive the same color is denoted by $\chi_{\bar{k}}(n)$. Equivalently, $\chi_{\bar{k}}(n)$ denotes the minimum number of binary codes with minimum distance at least $k+1$ required to partition the $n$-dimensional Hamming space. Using a computer search, we improve upon the known upper bound for $n=9$ by showing that $13 \leq \chi_{\bar{2}}(9) \leq 14$.