A robust Corradi-Hajnal theorem

For a graph G and p is an element of[0,1], we denote by G(p) the random sparsification of G obtained by keeping each edge of G independently, with probability p. We show that there exists a C>0 such that if p >= C(log n)(1/3)n(-2/3) and Gisann-vertex graph with n is an element of 3 N and delta(G) >= 2n/3, then with high probability G(p) contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of C, are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corradiand Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to p=1 in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in G(n,p) (corresponding to G=K-n in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least 2n/3 which gets close to the truth.