Counting and packing Hamilton `-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! · ( 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 Haimlton `-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.