Upper Bounds on the Minimum Size of Hamilton Saturated Hypergraphs

For 1 6 ` < k, an `-overlapping k-cycle is a k-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of k consecutive vertices and every two consecutive edges share exactly ` vertices. A k-uniform hypergraph H is `-Hamiltonian saturated if H does not contain an `-overlapping Hamiltonian k-cycle but every hypergraph obtained from H by adding one edge does contain such a cycle. Let sat(n, k, `) be the smallest number of edges in an `-Hamiltonian saturated k-uniform hypergraph on n vertices. In the case of graphs Clark and Entringer showed in 1983 that sat(n, 2, 1) = d 2 e. The present authors proved that for k > 3 and ` = 1, as well as for all 0.8k 6 ` 6 k− 1, sat(n, k, `) = Θ(n`). In this paper we prove two upper bounds which cover the remaining range of `. The first, quite technical one, restricted to ` > k+1 2 , implies in particular that for ` = 23k and ` = 3 4k we have sat(n, k, `) = O(n `+1). Our main result provides an upper bound sat(n, k, `) = O(n(k+`)/2) valid for all k and `. In the smallest open case we improve it further to sat(n, 4, 2) = O(n14/5).