Multipacking in Hypercubes

TL;DR

This paper investigates the relationship between broadcast domination and multipacking in hypercubes, proving bounds on multipacking numbers that support a conjecture relating these parameters and illustrating the ratio approaching 2.

Contribution

It establishes bounds on the multipacking number of hypercubes, confirming a conjecture and providing a sequence of graphs where the ratio of broadcast domination to multipacking approaches 2.

Findings

Bounds on multipacking number of hypercubes are proven.

The ratio of broadcast domination to multipacking approaches 2 in hypercubes.

Supports the conjecture that broadcast domination number is at most twice the multipacking number.

Abstract

For an undirected graph $G$, a dominating broadcast on $G$ is a function $f : V(G) \rightarrow \mathbb{N}$ such that for any vertex $u \in V(G)$, there exists a vertex $v \in V(G)$ with $f(v) \geqslant 1$ and $d(u,v) \leqslant f(v)$. The cost of $f$ is $\sum_{v \in V} f(v)$. The minimum cost over all the dominating broadcasts on $G$ is defined as the broadcast domination number $\gamma_b(G)$ of $G$. A multipacking in $G$ is a subset $M \subseteq V(G)$ such that, for every vertex $v \in V(G)$ and every positive integer $r$, the number of vertices in $M$ within distance $r$ of $v$ is at most $r$. The multipacking number of $G$, denoted $\operatorname{mp}(G)$, is the maximum cardinality of a multipacking in $G$. These two optimisation problems are duals of each other, and it easily follows that $\operatorname{mp}(G) \leqslant \gamma_b(G)$. It is known that $\gamma_b(G) \leqslant 2\operatorname{mp}(G)+3$ and conjectured that $\gamma_b(G) \leqslant 2\operatorname{mp}(G)$. In this paper, we show that for the $n$-dimensional hypercube $Q_n$ $$ \left\lfloor\frac{n}{2} \right\rfloor \leqslant \operatorname{mp}(Q_n) \leqslant \frac{n}{2} + 6\sqrt{2n}. $$ Since $\gamma_b(Q_n) = n-1$ for all $n \geqslant 3$, this verifies the above conjecture on hypercubes and, more interestingly, gives a sequence of connected graphs for which the ratio $\frac{\gamma_b(G)}{\operatorname{mp}(G)}$ approaches $2$, a search for which was initiated by Beaudou, Brewster and Foucaud in 2018. It follows that, for connected graphs $G$ $$ \limsup_{\operatorname{mp}(G) \rightarrow \infty} \left\{\frac{\gamma_b(G)}{\operatorname{mp}(G)}\right\} = 2.$$ The lower bound on $\operatorname{mp}(Q_n)$ is established by a recursive construction, and the upper bound is established using a classic result from discrepancy theory.