Positive co-degree thresholds for spanning structures

TL;DR

This paper establishes thresholds for the minimum positive co-degree in hypergraphs that guarantee the existence of spanning structures like Hamiltonian cycles and perfect matchings, advancing understanding of hypergraph spanning properties.

Contribution

It precisely determines the positive co-degree thresholds for Berge Hamiltonian cycles and asymptotically for loose Hamiltonian cycles in 3-graphs, and nearly exactly for perfect matchings.

Findings

Threshold for Berge Hamiltonian cycles in r-graphs determined.

Asymptotic threshold for loose Hamiltonian cycles in 3-graphs established.

Near-exact threshold for perfect matchings in all r-graphs provided.

Abstract

The \textit{minimum positive 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.