Weak and strong Lefschetz properties for vertex cover Artinian algebras associated to graphs

Let $G$ be a finite simple graph and let $A_c(G)$ be the Artinian algebra associated with its cover ideal. We prove that $A_c(G)$ has the WLP when $τ(G)>|V(G)|/2$, where $τ(G)$ denotes the size of a minimum vertex cover of $G$. As a consequence, $A_c(G)$ has the WLP with high probability when the Er