P\'osa-type results for Berge-hypergraphs

TL;DR

This paper establishes a sharp Pósa-type lower bound on the minimum degree conditions needed to guarantee Hamiltonian Berge cycles in both uniform and non-uniform hypergraphs, extending previous Dirac-type results.

Contribution

It introduces a new Pósa-type bound for hypergraphs that ensures the existence of Hamiltonian Berge cycles, advancing the understanding of degree conditions in hypergraph Hamiltonicity.

Findings

Provides a sharp Pósa-type lower bound for hypergraphs.

Extends Dirac-type conditions to Pósa-type bounds.

Applies to both uniform and non-uniform hypergraphs.

Abstract

A Berge cycle of length $k$ in a hypergraph $\mathcal H$ is a sequence of distinct vertices and hyperedges $v_1,h_1,v_2,h_2,\dots,v_{k},h_k$ such that $v_{i},v_{i+1}\in h_i$ for all $i\in[k]$, indices taken modulo $k$. F\"uredi, Kostochka and Luo recently gave sharp Dirac-type minimum degree conditions that force non-uniform hypergraphs to have Hamiltonian Berge cycles. We give a sharp P\'osa-type lower bound for $r$-uniform and non-uniform hypergraphs that force Hamiltonian Berge cycles.