2 problems

Co-diameter-two characterization conjecture. The graph $G$ is non-trivially minimally tough if and only if $G$ is isomorphic to $S_{\ell,\ell,\ell}$ for some $\ell\geq 1$.

Eliahou–Villarreal conjecture. If the edge ideal of a connected graph $G$ has a 2-linear resolution, then