Triangle Factors in Random Graphs

For a graph G = (V, E) on n vertices, where 3 divides n, a triangle factor is a subgraph of G, consisting of n/3 vertex disjoint triangles (complete graphs on three vertices). We discuss the problem of determining the minimal probability p = p(n), for which a random graph G ∈ script G;&(n, p) contains almost surely a triangle factor. This problem (in a more general setting) has been studied by Alon and Yuster and by Ruciński, their approach implies p = O((logn/n)^1/2). Our main result is that p = O(n^-3/5) already suffices. The proof is based on a multiple use of the Janson inequality. Our approach can be extended to improve known results about the threshold for the existence of an H-factor in script G(n, p) for various graphs H.