Solution to a problem on isolation of cliques in uniform hypergraphs

TL;DR

This paper establishes an upper bound on the minimum vertex set needed to isolate all complete r-uniform hypergraph copies in a connected hypergraph, solving a recent open problem and extending known graph results to hypergraphs.

Contribution

It generalizes the known graph case to r-uniform hypergraphs, providing bounds and characterizations for the isolation number in connected hypergraphs.

Findings

The isolation number is at most n/(k+1) for connected hypergraphs.

The result is tight except for specific extremal structures.

Extremal structures are characterized for all r-uniform hypergraphs.

Abstract

A copy of a hypergraph $F$ is called an $F$-copy. Let $K_k^r$ denote the complete $r$-uniform hypergraph whose vertex set is $[k] = \{1, \dots, k\}$ (that is, the edges of $K_k^r$ are the $r$-element subsets of $[k]$). Given an $r$-uniform $n$-vertex hypergraph $H$, the $K_k^r$-isolation number of $H$, denoted by $\iota(H, K_k^r)$, is the size of a smallest subset $D$ of the vertex set of $H$ such that the closed neighbourhood $N[D]$ of $D$ intersects the vertex sets of the $K_k^r$-copies contained by $H$ (equivalently, $H-N[D]$ contains no $K_k^r$-copy). In this note, we show that if $2 \leq r \leq k$ and $H$ is connected, then $\iota(H, K_k^r) \leq \frac{n}{k+1}$ unless $H$ is a $K_k^r$-copy or $k = r = 2$ and $H$ is a $5$-cycle. This solves a recent problem of Li, Zhang and Ye. The result for $r = 2$ (that is, $H$ is a graph) was proved by Fenech, Kaemawichanurat and the author, and is used to prove the result for any $r$. The extremal structures for $r = 2$ were determined by various authors. We use this to determine the extremal structures for any $r$.