Positive co-degree thresholds for spanning structures

The $\mathit{\text{minimum}}\mathit{\text{positive}}\mathit{\text{co-degree}}$ of a non-empty $r$-graph $H$, denoted ${\delta }_{r-1}^{+}(H)$, is the largest integer $k$ such that if a set $S\subset V(H)$ of size $r-1$ is contained in at least one $r$-edge of $H$, then $S$ is contained in at least $k$ $r$-edges of $H$. Motivated by several recent papers which study minimum positive co-degree as a reasonable notion of minimum degree in $r$-graphs, we consider bounds of ${\delta }_{r-1}^{+}(H)$ which will guarantee the existence of various spanning subgraphs in $H$. We precisely determine the minimum positive co-degree threshold for Berge Hamiltonian cycles in $r$-graphs, and asymptotically determine the minimum positive co-degree threshold for loose Hamiltonian cycles in $3$-graphs. For all $r$, we also determine up to an additive constant the minimum positive co-degree threshold for perfect matchings.