New Upper Bounds on the Minimal Domination Numbers of High-Dimensional Hypercubes

TL;DR

This paper introduces a new construction method for dominating sets in high-dimensional hypercubes, providing improved upper bounds on their minimal domination numbers within specific parameter ranges.

Contribution

The authors develop a novel construction technique for dominating sets in hypercubes that yields tighter upper bounds on the minimal domination number for certain dimensions.

Findings

New upper bounds for 4n improve previous results

Construction applies to a broad class of hypercube dimensions

Asymptotic analysis shows bounds are tighter in smaller wedges

Abstract

We briefly review known results on upper bounds for the minimal domination number $\gamma_n$ of a hypercube of dimension $n$, then present a new method for constructing dominating sets. Write $n =2^{\hat{n}}-1 +{\check{n}}$ with $0\leq {\check{n}}<2^{\hat{n}}$. Our construction applies to all $n$ lying within the expanding wedge $\theta({\hat{n}}) \leq {\check{n}} < 2^{{\hat{n}}}$, where $\theta$ is a specific, easily computable function with the asymptotic property $\theta(a) \sim 2^{a/2}$. For all $n$ within the smaller wedge $\theta({\hat{n}}) \leq {\check{n}} < 2^{{\hat{n}}-2}$, the resulting upper bound on $\gamma_n$ betters those previously known.