Berge pancyclic hypergraphs

An



n
n



-vertex graph



G
G



is
pancyclic
if for every




ℓ


\ell



,




3

≤



ℓ



≤


n

3\leq \ell \leq n



,



G
G



contains a cycle of length




ℓ


\ell



. We consider an analog for hypergraphs. An



r
r



-uniform hypergraph is a set of vertices and a set of edges in which every edge contains



r
r



vertices. A Berge cycle of length




ℓ


\ell



in a hypergraph is an alternating sequence of




ℓ


\ell



distinct vertices and




ℓ


\ell



distinct edges





v
1

,

e
1

,

v
2

,

…


,

v

ℓ



,

e


ℓ




,

v
1


v_1,e_1,v_2, \ldots , v_\ell , e_{\ell },v_1



such that




{

v
i

,

v

i
+
1


}

⊆



e
i


\{v_i, v_{i+1}\} \subseteq e_i



for all



i
i



, with indices taken modulo




ℓ


\ell



. We similarly call a hypergraph
Berge pancyclic
if it contains Berge cycles of every possible length.


For sufficiently large



n
n



we prove a sharp minimum degree condition guaranteeing Berge pancyclicity in



r
r



-uniform hypergraphs where




3

≤


r

≤



3 \leq r \leq








⌊


(
n

−


1
)

/

2

⌋



−


2

\lfloor (n-1)/2\rfloor - 2



. This bound is optimal for all such combinations of



n
n



and



r
r



.