Dirac-type results for loose Hamilton cycles in uniform hypergraphs

A classic result of G. A. Dirac in graph theory asserts that every n-vertex graph (n ≥ 3) 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. Rödl, A. Ruciński, and E. Szemerédi 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 `-cycles. A Hamilton `-cycle in an n-vertex, k-uniform hypergraph (1 ≤ ` < 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 ` vertices. We prove sufficient minimum (k − 1)-degree conditions for Hamilton `cycles if ` < k/2. In particular, we show that for every ` < k/2 every n-vertex, k-uniform hypergraph with minimum (k−1)-degree (1/(2(k−`))+o(1))n contains such a loose Hamilton `-cycle. This degree condition is approximately tight and was conjectured by D. Kühn and D. Osthus (for ` = 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. Rödl, A. Ruciński, and E. Sze-