Perfect matchings in random uniform hypergraphs

Let ℋk(n, p) be the random k-uniform hypergraph on V = [n] with edge probability p. Motivated by a theorem of Erdős and Renyi 7 regarding when a random graph G(n, p) = ℋ2(n, p) has a perfect matching, the following conjecture may be raised. (See J. Schmidt and E. Shamir 16 for a weaker version.) Conjecture. Let k|n for fixed k ≥ 3, and the expected degree d(n, p) = p(). Then (Erdős and Renyi 7 proved this for G(n, p).) Assuming d(n, p)/n1/2 ∞, Schmidt and Shamir 16 were able to prove that ℋk(n, p) contains a perfect matching with probability 1 − o(1). Frieze and Janson 8 showed that a weaker condition d(n, p)/n1/3 ∞ was enough. In this paper, we further weaken the condition to A condition for a similar problem about a perfect triangle packing of G(n, p) is also obtained. A perfect triangle packing of a graph is a collection of vertex disjoint triangles whose union is the entire vertex set. Improving a condition p ≥ cn−2/3+1/15 of Krivelevich 12, it is shown that if 3|n and p ≫ n−2/3+1/18, then © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 111–132, 2003