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Solution to a problem on isolation of cliques in uniform hypergraphs

Peter Borg

TL;DR

This work establishes a tight upper bound on the $K_k^r$-isolation number for connected $r$-uniform hypergraphs with $2 \le r \le k$, showing $\iota(H, K_k^r) \le \frac{n}{k+1}$ except when $H$ is a copy of $K_k^r$ or $(k,r)=(2,2)$ with $H$ a $5$-cycle. The main strategy reduces the hypergraph problem to the graph case by analyzing the $2$-shadow $H^{(2)}$ and applying the known graph result, then lifting the bound back to $H$; equality is achieved precisely by pure $(n,K_k^r)$-good $r$-graphs, or, for small $k$, by specific exceptional hypergraph families. The paper further delineates the extremal structures for $r=2$ and extends them to all $r$, yielding a complete description of when the bound is tight. It also discusses the more delicate case $k<r$, linking isolation to domination and presenting open problems and constructions that demonstrate sharpness of related bounds. Overall, the work extends graph isolation results to hypergraphs and clarifies the extremal configurations that attain the bound.

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 $ι(H, K_k^r)$, 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 $ι(H, K_k^r) \leq \frac{n}{k+1}$ 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 $r$.

Solution to a problem on isolation of cliques in uniform hypergraphs

TL;DR

This work establishes a tight upper bound on the -isolation number for connected -uniform hypergraphs with , showing except when is a copy of or with a -cycle. The main strategy reduces the hypergraph problem to the graph case by analyzing the -shadow and applying the known graph result, then lifting the bound back to ; equality is achieved precisely by pure -good -graphs, or, for small , by specific exceptional hypergraph families. The paper further delineates the extremal structures for and extends them to all , yielding a complete description of when the bound is tight. It also discusses the more delicate case , linking isolation to domination and presenting open problems and constructions that demonstrate sharpness of related bounds. Overall, the work extends graph isolation results to hypergraphs and clarifies the extremal configurations that attain the bound.

Abstract

A copy of a hypergraph is called an -copy. Let denote the complete -uniform hypergraph whose vertex set is (that is, the edges of are the -element subsets of ). Given an -uniform -vertex hypergraph , the -isolation number of , denoted by , is the size of a smallest subset of the vertex set of such that the closed neighbourhood of intersects the vertex sets of the -copies contained by (equivalently, contains no -copy). In this note, we show that if and is connected, then unless is a -copy or and is a -cycle. This solves a recent problem of Li, Zhang and Ye. The result for (that is, is a graph) was proved by Fenech, Kaemawichanurat and the author, and is used to prove the result for any . The extremal structures for were determined by various authors. We use this to determine the extremal structures for any .
Paper Structure (3 sections, 6 theorems, 4 equations)

This paper contains 3 sections, 6 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. The case $k < r$

Key Result

Theorem 1

If $k \geq 1$ and $G$ is a connected $n$-vertex graph, then, unless either $G \simeq K_k$ or $k = 2$ and $G \simeq C_5$, Moreover, there exists a graph $B_{n,k}$ such that $\iota(B_{n,k}, K_k) = \lfloor n/(k+1) \rfloor$.

Theorems & Definitions (6)

  • Theorem 1: BFK
  • Theorem 2
  • Theorem 3: CCZCCZ2
  • Theorem 4
  • Lemma 1
  • Theorem 5