Berge Pancyclic hypergraphs

TL;DR

This paper establishes a sharp minimum degree condition for uniform hypergraphs to contain Berge cycles of all lengths from 3 to n, thus guaranteeing pancyclicity.

Contribution

It proves a new Dirac-type theorem that characterizes when an r-uniform hypergraph is pancyclic based on minimum degree thresholds.

Findings

Hypergraphs with minimum degree above a specific threshold are pancyclic.

The result is sharp for large n and certain uniformities r.

Provides a new criterion for cycle existence in hypergraphs.

Abstract

A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2, \ldots, v_\ell, e_{\ell}$ such that $\{v_i, v_{i+1}\} \subseteq e_i$ for all $i$, with indices taken modulo $\ell$. We call an $n$-vertex hypergraph pancyclic if it contains Berge cycles of every length from $3$ to $n$. We prove a sharp Dirac-type result guaranteeing pancyclicity in uniform hypergraphs: for $n \geq 70$, $3 \leq r \leq \lfloor (n-1)/2\rfloor - 2$, if $\cH$ is an $n$-vertex, $r$-uniform hypergraph with minimum degree at least ${\lfloor (n-1)/2 \rfloor \choose r-1} + 1$, then $\cH$ is pancyclic.