论文标题
$ k $ power的统治号码的上限$ r $ - 均匀的超图
An upper bound for the $k$-power domination number in $r$-uniform hypergraphs
论文作者
论文摘要
在图形上概括工作，Chang和Roussel引入了$ K $ - 功率统治超图中的统治，并猜想了$ r $ r $ r $ rustriform Hypergraphs $ n $ Vertices的上限为$ \ frac {n} {n} {r+k} $。对于简单的图表（$ r = 2 $），该上限被证明是正确的，并且进一步猜想只有一个被称为鱿鱼超图的超图系列才能获得此上限。在本文中，该猜想被证明可以用于$ r = 3 $或$ 4 $的超图；但通过反例显示为$ r \ geq 7 $。此外，我们表明鱿鱼超图并不是唯一获得原始上限的超图。最后，新的上限被证明是$ r \ geq 3 $。
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 $r\geq 3$.
