Minimum degree ensuring that a hypergraph is hamiltonian-connected

TL;DR

This paper establishes exact minimum degree thresholds for r-uniform hypergraphs to be hamiltonian-connected, revealing differences from graph cases and constructing hypergraphs that are hamiltonian-connected without hamiltonian Berge cycles.

Contribution

It provides the first exact degree bounds for hamiltonian-connectedness in hypergraphs and highlights unique properties distinct from graphs.

Findings

Exact lower bounds for minimum degree ensuring hamiltonian-connectedness.

For certain r, hypergraphs can be hamiltonian-connected without hamiltonian Berge cycles.

The degree threshold is just below that for hamiltonian Berge cycle existence.

Abstract

A hypergraph $H$ is hamiltonian-connected if for any distinct vertices $x$ and $y$, $H$ contains a hamiltonian Berge path from $x$ to $y$. We find for all $3\leq r<n$, exact lower bounds on minimum degree $\delta(n,r)$ of an $n$-vertex $r$-uniform hypergraph $H$ guaranteeing that $H$ is hamiltonian-connected. It turns out that for $3\leq n/2<r<n$, $\delta(n,r)$ is 1 less than the degree bound guaranteeing the existence of a hamiltonian Berge cycle. Moreover, unlike for graphs, for each $r \geq 3$ there exists an $r$-uniform hypergraph that is hamiltonian-connected but does not contain a hamiltonian Berge cycle.