Hamilton ℓ-cycles in uniform hypergraphs

Abstract

We say that a k-uniform hypergraph C is an ℓ- cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ℓ vertices. We prove that if 1 ⩽ ℓ < k and k − ℓ does not divide k then any k-uniform hypergraph on n vertices with minimum degree at least n ⌈ k k − ℓ ⌉ ( k − ℓ ) + o ( n ) contains a Hamilton ℓ-cycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an ℓ-cycle for any ℓ with 1 ⩽ ℓ < k .