of all monomial ideals of $S$ generated in degree two and $\MT$ be the set of all graphs on the vertex set $[n]$ which do not have double edges but may have loops. There is an obvious bijection between $\MM$ and $\MT$. Indeed, if $I\in \MM$ then the graph $G\in \MT$ corresponding to $I$ has the edge set $E(G)=\{\{i,j\}\: x_ix_j\in G(I)\}$. In case $i=j$, the corresponding edge is a loop.

We say a graph $G$ is gap free if for any two disjoint edges $e,e'\in E(G)$ there exists an edge $f\in E(G)$ such that $e\cap f\neq\emptyset\neq e'\cap f$. In the case that $G$ is simple, $G$ is gap free if and only if its complement $\bar{G}$ has no 4-cycle.

\[main\] Let $G$ be a finite graph (possibly with loops) and let $I$ be its edge ideal. The following conditions are equivalent:

-   $G$ is gap free.

-   $\index (I^{k})>1$ for all $k\geq 1$.

-   $\index(I^{k})>1$ for some $k\geq 1$.

(a)(b): By Corollary \[connected\] it is enough to show that for all $k\geq 1$ and for all $u, v\in G(I^k)$ there is a path in $G^{(u,v)}_{I^k}$ connecting $u$ and $v$. We may assume that $\deg(\lcm(u,v))>2k+1$, since otherwise the assertion is trivial. 