Counting and packing Hamilton ℓ-cycles in dense hypergraphs

A k-uniform hypergraph H contains a Hamilton -cycle, if there is a cyclic ordering of the vertices of H such that the edges of the cycle are segments of length k in this ordering and any two consecutive edges fi, fi+1 share exactly vertices. We consider problems about packing and counting Hamilton -cycles in hypergraphs of large minimum degree. Given a hypergraph H, for a d-subset A ⊆ V (H), we denote by dH(A) the number of distinct edges f ∈ E(H) for which A ⊆ f, and set δd(H) to be the minimum dH(A) over all A ⊆ V (H) of size d. We show that if a k-uniform hypergraph on n vertices H satisfies δk−1(H) ≥ αn for some α > 1/2, then for every < k/2 H contains (1−o(1))n ·n!·( α!(k−2)!) n k− Hamilton -cycles.
The exponent above is easily seen to be optimal. In addition, we show that if δk−1(H) ≥ αn for α > 1/2, then H contains f(α)n edge-disjoint Hamilton -cycles for an explicit function f(α) > 0.
For the case where every (k−1)-tuple X ⊂ V (H) satisfies dH(X) ∈ (α ± o(1))n, we show that H contains edge-disjoint Hamilton - cycles which cover all but o(|E(H)|) edges of H. As a tool we prove the following result which might be of independent interest: For a bipartite graph G with both parts of size n, with minimum degree
at least δn, where δ > 1/2, and for p = ω(log n/n) the following holds. If G contains an r-factor for r = Θ(n), then by retaining edges of G with probability p independently at random, w.h.p the resulting graph contains a (1 − o(1))rp-factor.