A graph $G$ is called a $2K_2$-free graph if it does not contain $2K_2$ as an induced subgraph. In 2014, Broersma, Patel and Pyatkin showed that every 25-tough $2K_2$-free graph on at least three vertices is Hamiltonian. Recently, Shan improved this result by showing that 3-tough is sufficient instead of 25-tough. In this paper, we show that every 2-tough $2K_2$-free graph on at least three vertices is Hamiltonian, which was conjectured by Gao and Pasechnik.

14 pages. We have noticed that one of our results, showing that every $\frac{3}{2}$-tough $2K_2$-free graph on at least three vertices has a 2-factor, is an immediate corollary to a known result. So, we have just described the result on 2-factors as a proposition, and have omitted our proof (Section 3 in the previous version) in the paper. Accordingly, we have changed the title. $ \ $