A subset $M$ of the edges of a graph or hypergraph is hitting if $M$ covers each vertex of $H$ at least once, and $M$ is $t$-shallow if it covers each vertex of $H$ at most $t$ times. We consider the existence of shallow hitting edge sets and the maximum size of shallow edge sets in $r$-uniform hypergraph $H$ that are regular or have a large minimum degree. Specifically, we show the following. Every $r$-uniform regular hypergraph has a $t$-shallow hitting edge set with $t = O(r)$. Every $r$-uniform regular hypergraph with $n$ vertices has a $t$-shallow edge set of size $\Omega(nt/r^{1+1/t})$. Every $r$-uniform hypergraph with $n$ vertices and minimum degree $\delta_{r-1}(H) \geq n/((r-1)t+1)$ has a $t$-shallow hitting edge set. Every $r$-uniform $r$-partite hypergraph with $n$ vertices in each part and minimum degree $\delta'_{r-1}(H) \geq n/((r-1)t+1) +1$ has a $t$-shallow hitting edge set. We complement our results with constructions of $r$-uniform hypergraphs that show that most of our obtained bounds are best-possible.

翻译：图或超图边的子集$M$称为覆盖集，若$M$覆盖$H$的每个顶点至少一次；$M$称为$t$-浅层覆盖集，若其覆盖$H$的每个顶点至多$t$次。本文研究正则或具有大最小度的$r$-均匀超图$H$中浅层边覆盖集的存在性及浅层边集的最大规模。具体结果如下：每个$r$-均匀正则超图均存在$t$-浅层覆盖集，其中$t = O(r)$；每个具有$n$顶点的$r$-均匀正则超图存在规模为$\Omega(nt/r^{1+1/t})$的$t$-浅层边集；每个具有$n$顶点且最小度$\delta_{r-1}(H) \geq n/((r-1)t+1)$的$r$-均匀超图存在$t$-浅层覆盖集；每个各部分具有$n$顶点且最小度$\delta'_{r-1}(H) \geq n/((r-1)t+1) +1$的$r$-均匀$r$-部超图存在$t$-浅层覆盖集。我们通过构造$r$-均匀超图实例，证明所得界值大多为最优的。