Berge Pancyclic hypergraphs

Abstract

A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2,\dots ,v_{\ell },e_{\ell }$ 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 $n\ge 70$, $3\le r\le \lfloor (n-1)/2\rfloor -2$, if $\cH$ is an $n$-vertex, $r$-uniform hypergraph with minimum degree at least $\left(\genfrac{}{}{0ex}{}{\lfloor (n-1)/2\rfloor }{r-1}\right)+1$, then $\cH$ is pancyclic.