An approximate ore-type result for tight Hamilton cycles in uniform hypergraphs

A classic result of O. Ore in 1960 is that if the degree sum of any two independent vertices in an n-vertex graph is at least n, then the graph contains a Hamiltonian cycle. Naturally, we consider to generalize it to hypergraphs situation. In this paper, we prove the following theorems. (i) For any n≥4k2, there is an n-vertex k-uniform hypergraph, with degree sum of any two strongly independent sets of k−1 vertices bigger than 2n−4(k−1), contains no Hamilton l-cycle, 1≤ℓ≤k−1. (ii) If the degree sum of two weakly independent sets of k−1 vertices in an n-vertex k-uniform hypergraph is (1+o(1))n, then the hypergraph contains a Hamilton (k−1)-cycle, where two distinct sets of k−1 vertices are weakly (strongly) independent if there exist no edge containing the union of them (intersecting both of them).