Hamilton-chain saturated hypergraphs

Abstract

We say that a hypergraph H is hamiltonian path (cycle) saturated if H does not contain an open (closed) hamiltonian chain but by adding any new edge we create an open (closed) hamiltonian chain in H . In this paper we ask about the smallest size of an r -uniform hamiltonian path (cycle) saturated hypergraph, mainly for r = 3 . We present a construction of a family of 3-uniform path (cycle) saturated hamiltonian hypergraphs with O ( n 5 / 2 ) edges. On the other hand we prove that the number of edges in an r -uniform hamiltonian path (cycle) saturated hypergraph is at least Ω ( n r − 1 ) .