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

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.