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Stability results for Berge-matching in hypergraphs

Jia-Bao Yang, Leilei Zhang

TL;DR

The paper proves stability results for Berge-matchings in $r$-uniform hypergraphs by translating Berge-$F$-free hypergraphs into red-blue graphs and applying a Tutte–Berge-based decomposition. A key technical lemma bounds the maximum gain $g_r(G)$ over red-blue graphs associated with the extremal hypergraph $H(n,k,t,c)$, enabling a structural description: any near-extremal Berge-$M_{k+1}$-free hypergraph must be close to subgraphs of the $H(n,k,t,c)$ families, with $t$ restricted to certain ranges. Consequently, the authors derive stability versions of the Erdős–Gallai theorem for matchings, and parallel stability statements for the hypergraph Berge setting, including precise corollaries for large $n$ that place near-extremal graphs within the known extremal constructions. These results deepen the understanding of how near-extremal configurations must resemble the extremal ones and extend classical stability phenomena from graphs to Berge-hypergraphs.

Abstract

Given a graph $F$, a hypergraph is called a Berge-$F$ if it can be obtained by expanding each edge of $F$ into a hyperedge containing it. Let $M_{k}$ denote the matching of size $k$. Kang, Ni, and Shan [12] determined the Turán number of Berge-$M_k$. Our main result shows that if an $r$-uniform hypergraph $H$ on $n$ vertices has nearly as many edges as the extremal in their theorem without containing $M_k$, then $H$ must be structurally close to certain well-specified graphs. Meanwhile, our result also implies several stability results, such as the stability version of the well-known Erdős-Gallai theorem (Erdős and Gallai, 1959 [5]).

Stability results for Berge-matching in hypergraphs

TL;DR

The paper proves stability results for Berge-matchings in -uniform hypergraphs by translating Berge--free hypergraphs into red-blue graphs and applying a Tutte–Berge-based decomposition. A key technical lemma bounds the maximum gain over red-blue graphs associated with the extremal hypergraph , enabling a structural description: any near-extremal Berge--free hypergraph must be close to subgraphs of the families, with restricted to certain ranges. Consequently, the authors derive stability versions of the Erdős–Gallai theorem for matchings, and parallel stability statements for the hypergraph Berge setting, including precise corollaries for large that place near-extremal graphs within the known extremal constructions. These results deepen the understanding of how near-extremal configurations must resemble the extremal ones and extend classical stability phenomena from graphs to Berge-hypergraphs.

Abstract

Given a graph , a hypergraph is called a Berge- if it can be obtained by expanding each edge of into a hyperedge containing it. Let denote the matching of size . Kang, Ni, and Shan [12] determined the Turán number of Berge-. Our main result shows that if an -uniform hypergraph on vertices has nearly as many edges as the extremal in their theorem without containing , then must be structurally close to certain well-specified graphs. Meanwhile, our result also implies several stability results, such as the stability version of the well-known Erdős-Gallai theorem (Erdős and Gallai, 1959 [5]).
Paper Structure (4 sections, 12 theorems, 77 equations, 2 figures)

This paper contains 4 sections, 12 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of Theorem \ref{['stabilitytheorem-for-bergematching']}
  4. Proofs of Theorem \ref{['stabilitytheorem-for-bergematching-sufficiently']} and Corollary \ref{['corollary']}

Key Result

Theorem 1.1

Let $n$ and $k$ be integers with $n\geq 2k+1$. Then

Figures (2)

  • Figure 1: $H(13,6,3,(5,3,1,1))$
  • Figure 2: $G^{\prime} \to G_{1}$

Theorems & Definitions (14)

  • Theorem 1.1: Erdős and Gallai Erdos1959
  • Conjecture 1.2: Erdős Erdos1965
  • Theorem 1.3: Kang, Ni, and Shan Kang2022
  • Theorem 1.4: Stability
  • Corollary 1.5
  • Theorem 1.6: Stability
  • Corollary 1.7
  • Theorem 2.1: Tutte-Berge Theorem Lovász2009
  • Lemma 2.2: Gerbner, Methuku, and Palmer Gerbnerm2019
  • Lemma 2.3: Kang, Ni, and Shan Kang2022
  • ...and 4 more