An upper bound for the $k$-power domination number in $r$-uniform hypergraphs

TL;DR

This paper investigates the upper bounds of the $k$-power domination number in $r$-uniform hypergraphs, proving the conjecture for certain cases, providing counterexamples for others, and establishing a new bound.

Contribution

It confirms the conjectured upper bound for $r=3,4$, disproves it for $r extgreater=7$, and introduces a new upper bound for all $r extgreater=3$.

Findings

Conjecture holds for $r=3,4$ hypergraphs.

Counterexample disproves the conjecture for $r extgreater=7$.

New upper bound established for $r extgreater=3$.

Abstract

Generalizing work on graphs, Chang and Roussel introduced $k$-power domination in hypergraphs and conjectured the upper bound for the $k$-power domination number for $r$-uniform hypergraphs on $n$ vertices was $\frac{n}{r+k}$. This upper bound was shown to be true for simple graphs ($r=2$) and it was further conjectured that only a family of hypergraphs, known as the squid hypergraphs, attained this upper bound. In this paper, the conjecture is proven to hold for hypergraphs with $r=3$ or $4$; but is shown to be false, by a counterexample, for $r\geq 7$. Furthermore, we show that the squid hypergraphs are not the only hypergraphs that attain the original upper bound. Finally, a new upper bound is proven for $r\geq 3$.