Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree

We say that a 3-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. Also, let C4 denote the 3-uniform hypergraph on 4 vertices which contains 2 edges. We prove that for every ε > 0 there is an n0 such that for every n n0 the following holds: Every 3-uniform hypergraph on n vertices whose minimum degree is at least n/4+ εn contains a Hamilton cycle. Moreover, it also contains a perfect C4-packing. Both these results are best possible up to the error term εn. © 2006 Elsevier Inc. All rights reserved.