for large powers (see [@B], [@Ca], [@Ch], [@CHT], [@Co], [@HH], [@HW], [@HHZ1], [@HHZ2]), while their initial behavior is often quite mysterious, even for monomial ideals. However with many respects monomial ideals generated in degree 2 behave more controllable from the very beginning. So now let $I$ be a monomial ideal generated in degree $2$. The second author together with Hibi and Zheng showed in [@HHZ2] that if $I$ has a linear resolution, then all of its powers have a linear resolution as well. More recently there have been several interesting generalizations of this result. In case that $I$ is squarefree, $I$ may be viewed as the edge ideal of a finite simple graph $G$, and in this case Francisco, H$\grave{a}$ and Van Tuyl raised the question whether $I^k$ has a linear resolution for $k\geq 2$, assuming the complementary graph contains no induced $4$-cycle, equivalently, $G$ is gap free. However, Nevo and Peeva showed by an example [@NP Counterexample 1.10] that this is not always the case. On the other hand, Nevo [@N] showed that $I^2$ has a linear resolution if $G$ is gap and claw free, and Banerjee [@B] gives a positive answer to the above question under the additional 