Dirac-type results for loose Hamilton cycles in uniform hypergraphs

Abstract

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. 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 ℓ-cycles. A Hamilton ℓ-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 ℓ 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 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. Kühn 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. Rödl, A. Ruciński, and E. Szemerédi.