Counting and packing Hamilton L-cycles in dense hypergraphs

A k-uniform hypergraph H contains a Hamilton f-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 f(i), f(i)+1 share exactly L vertices. We consider problems about packing and counting Hamilton f-cycles in hypergraphs of large minimum degree. Given a hypergraph H, for a d-subset A subset of V(H), we denote by d(H)(A) the number of distinct edges f is an element of E(H) for which A subset of f, and set (5d(H) to be the minimum djj(A) over all A subset of V(H) of size d. We show that if a k-uniform hypergraph on n vertices H satisfies (delta(k-1)(H) > an for some a > 1/2, then for every l < k/2 H contains (1-o(1))(n).n! (alpha/l!(k-2l))(n/k-l) Hamilton f-cycles. The exponent above is easily seen to be optimal. In addition, we show that if (delta(k-1)(H) >= an for alpha > 1/2, then H contains f(alpha)n edge-disjoint Hamilton f-cycles for an explicit function f(alpha) > 0. For the case where every (k-1)-tuple X C V(H) satisfies dH(X) is an element of (alpha +/- o(1))(n), we show that H contains edge-disjoint Hamilton f' cycles which cover all but o(vertical bar E(H)vertical bar) 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 delta n, where delta > 1/2, and for p = omega(logn/n) the following holds. If G contains an r-factor for r = Theta(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.