Dirac-type results for loose Hamilton cycles in uniform hypergraphs

A classic result of G.A. Dirac in graph theory asserts that for n >= 3 every n-vertex graph with minimum degree at least n/2 contains a spanning (so-called Hamilton) cycle. G.Y. Katona and H.A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutive (modulo n) vertices in the ordering form an edge. Moreover. the minimum degree is the minimum (k - 1)degree, i.e. the minimum number of edges containing a fixed set of k - 1 vertices. V. Rodl, A. Rucinski, and E. Szemeredi verified (approximately) the conjecture of Katona and Kierstead and showed that every n-vertex, k-uniform hypergraph with minimum (k - 1)-degree (1/2 + o(1))n contains such a tight Hamilton cycle. We study the similar question for Hamilton l-cycles. A Hamilton e-cycle in an n-vertex, k-Uniform hypergraph (1 <= l < k) is an ordering of the vertices and an ordered subset of the edges such that each such edge corresponds to k consecutive (modulo n) vertices and two consecutive edges intersect in precisely e vertices. We prove sufficient minimum (k - 1)-degree conditions for Hamilton l-cycles if l < k/2. In particular, we show that for every l < k/2 every l < k/1 every n-vertex, k-uniform hypergraph with minimum (k -1)-degree (1/(2(k - l)) +o(1))n contains such a loose Hamilton l-cycle. This degree condition is approximately tight and was conjectured by D. Kuhn and D. Osthus (for l = 1). who verified it when k = 3. Our proof is based on the so-called weak regularity lemma for hypergraphs and follows the approach of V. Rodl, A. Rucinski, and E. Szemeredi. (C) 2009 Elsevier Inc. All rights reserved.