A robust Corrádi–Hajnal theorem

Abstract

For a graph and , we denote by the random sparsification of obtained by keeping each edge of independently, with probability . We show that there exists a such that if and is an ‐vertex graph with and , then with high probability contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of , are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corrádi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in (corresponding to in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least which gets close to the truth.