Stability results for Berge-matching in hypergraphs

TL;DR

This paper establishes stability results for Berge-matchings in hypergraphs, showing that hypergraphs nearly extremal in edge count are structurally close to specific configurations, extending classical theorems like Erdős-Gallai.

Contribution

It provides the first stability results for Berge-matchings in hypergraphs, linking near-extremal edge counts to structural proximity to specific graphs.

Findings

Hypergraphs with nearly maximum edges are structurally close to extremal configurations.

The results imply stability versions of classical theorems like Erdős-Gallai.

The paper extends stability concepts to Berge-matchings in 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 Tur\'an 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 Erd\H{o}s-Gallai theorem (Erd\H{o}s and Gallai, 1959 [5]).