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

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 nr+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≥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≥3.