An Ore-type theorem for $[3]$-graphs

Ore’s Theorem states that if $G$ is an $n$-vertex graph and every pair of non-adjacent vertices has degree sum at least $n$, then $G$ is Hamiltonian. A $[3]$-graph is a hypergraph in which every edge contains at most $3$ vertices. In this paper, we prove an Ore-type result on the existence of Hamiltonian Berge cycles in $[3]$-graph $\text{cH}$, based on the degree sum of every pair of non-adjacent vertices in the $2$-shadow graph $\mathrm{\partial }\text{cH}$ of $\text{cH}$. Namely, we prove that there exists a constant $d_0$ such that for all $n\ge 6$, if a $[3]$-graph $\text{cH}$ on $n$ vertices satisfies that every pair $u,v\in V(\text{cH})$ of non-adjacent vertices has degree sum $d_{\mathrm{\partial }\text{cH}}(u)+d_{\mathrm{\partial }\text{cH}}(v)\ge n+d_0$, then $\text{cH}$ contains a Hamiltonian Berge cycle. Moreover, we conjecture that $d_0=1$ suffices.