  Let $G$ be a $(C_4, 2K_2)$-free graph with edge ideal $I(G)\subset
\Bbbk[x_1,\dots , x_n]$. We show that $I(G)^s$ has linear resolution for every
$s\geq 2$. Also, we show that every power of the vertex cover ideal of $G$ has
linear quotients. As a result, we describe the Castelnuovo-Mumford regularity
of powers of $I(G)^{\vee}$ in terms of the maximum degree of $G$.
