Primality and complete-intersection conjecture for graph ideals

Let $d\geq 1$, let $K$ be an arbitrary field, and let $G$ be a graph on vertex set $[n]$. Primality and complete-intersection conjecture. If $G$ is $(n-d)$-connected, then $L_G$ is prime and a complete intersection. For $d=2$, the corresponding result is known in the characteristic hypotheses stated in the paper; the general assertion is presented as suggested by experimental data and remains open.