On Hamiltonian Berge cycles in $3$-uniform hypergraphs

Given a set $R$, a hypergraph is $R$-uniform if the size of every hyperedge belongs to $R$. A hypergraph $\mathcal{H}$ is called $\mathit{\text{covering}}$ if every vertex pair is contained in some hyperedge in $\mathcal{H}$. In this note, we show that every covering $[3]$-uniform hypergraph on $n\ge 6$ vertices contains a Berge cycle $C_s$ for any $3\le s\le n$. As an application, we determine the maximum Lagrangian of $k$-uniform Berge-$C_t$-free hypergraphs and Berge-$P_t$-free hypergraphs.