On the connected Tur\'an number of Berge paths and Berge cycles

TL;DR

This paper determines the exact connected Turán number for Berge paths and cycles in hypergraphs, resolving previous open questions and improving known bounds by reducing the threshold for extremal configurations.

Contribution

It proves the extremal number holds for all k ≥ 2r+2 and fails for k ≤ 2r+1, fully resolving a problem posed by Győri, Salia, and Zamora, and improves related results.

Findings

Extremal number holds for all k ≥ 2r+2

Extremal number fails for all k ≤ 2r+1

Reduces Berge-Turán problems to classical extremal graph theory

Abstract

Given a graph $F$, a Berge copy of $F$ (Berge-$F$ for short) is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H{o}ri, Salia and Zamora determined the maximum number of hyperedges in a connected $r$-uniform hypergraph on $n$ vertices containing no Berge path of length $k-1$ for all $k\geq 2r+14$ and sufficiently large $n$, and asked for the minimum $k_0$ such that this extremal number holds for all $k\geq k_0$. In this paper, we prove that the extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$, thereby completely resolving the problem posed by Gy\H{o}ri, Salia and Zamora. Moreover, we improve the result of F\"uredi, Kostochka and Luo, who determined the maximum number of hyperedges in a $2$-connected $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$ for all $k\geq 4r$ and sufficiently large $n$, by showing that this extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$. Our approach reduces Berge-Tur\'an problems to classical extremal graph theory problems, and applies recent work of Ai, Lei, Ning and Shi concerning the feasibility of graph parameters and the Kelmans operation.