Minimum degree ensuring that a hypergraph is hamiltonian-connected

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≤r<n, exact lower bounds on minimum degree δ(n,r) of an n-vertex r-uniform hypergraph H guaranteeing that H is hamiltonian-connected. It turns out that for 3≤n/2<r<n, δ(n,r) is 1 less than the degree bound guaranteeing the existence of a hamiltonian Berge cycle. Moreover, unlike for graphs, for each r≥3 there exists an r-uniform hypergraph that is hamiltonian-connected but does not contain a hamiltonian Berge cycle.