Saturation numbers for $3$-uniform Berge-$K_4$

TL;DR

This work determines the exact 3-uniform Berge-$K_4$-saturation number $ ext{sat}_3(n, ext{Berge-}K_4)$ for $n=5,7,8$ and all $n\ge 96$, with a special case $ ext{sat}_3(6, ext{Berge-}K_4)=5$. The authors combine two explicit upper-bound constructions, based on a tight $5$-cycle ${\mathcal C}^3_5$ and a gadget ${\mathcal T}$, with a rigorous lower-bound analysis that uses a vertex-partition and a careful study of edge-types between parts, to pin down the threshold at which the linear bound $n$ is achieved. They also classify all extremal hypergraphs for $5\le n\le 8$ via computer search and demonstrate the existence of many non-isomorphic extremal families for large $n$, suggesting a robust recursive structure via ${\mathcal T}$-augmentation. The paper develops an algorithmic framework for generating Berge-$K_\ell$-saturated hypergraphs, enabling broader exploration of saturation numbers in hypergraphs and offering concrete tools for future research.

Abstract

The saturation number $\text{sat}_r(n,\mathcal{F})$ is the minimum number of hyperedges in an $r$-uniform $\mathcal{F}$-saturated hypergraph on $n$ vertices. We determine this parameter for $3$-uniform Berge-$K_4$ hypergraphs, proving that $\text{sat}_3(n,\text{Berge-}K_4)=n$ for $n =5,7,8$ and $n\ge 96$, while $\text{sat}_3(6,\text{Berge-}K_4)=5$. This resolves a problem posed by English, Kritschgau, Nahvi, and Sprangel~\cite{EKNS2024} for large $n.$ Using a computer search, we classify all extremal hypergraphs for $5\le n\le 8.$ For $n\geq 96$, we further show the existence of many non-isomorphic extremal families. Our approach synthesizes structural insights with computational power.

Figures (2)

• Figure 1: Figure of Construction \ref{['Construction: odd n']}
• Figure 2: Figure of Construction \ref{['Construction: even n']}

Theorems & Definitions (18)

• Theorem 1.1
• Lemma 2.1
• proof
• Claim 2.2
• proof : Proof of Claim \ref{['ref:Claim u v adjacent']}
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof