Shallow Hitting Edge Sets in Uniform Hypergraphs

TL;DR

This paper investigates the existence and size of shallow hitting edge sets in uniform hypergraphs, providing bounds and constructions that are close to optimal, with implications for hypergraph covering problems.

Contribution

It establishes new bounds for shallow hitting edge sets in regular and degree-regular hypergraphs, and demonstrates the near-optimality of these bounds through constructions.

Findings

Every r-uniform regular hypergraph has a t-shallow hitting edge set with t = O(r)

Existence of large t-shallow edge sets in regular hypergraphs with size bounds

Most bounds are shown to be tight through hypergraph constructions

Abstract

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.