Ratliff Property of Edge Ideals of Weighted Oriented Graphs

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. In this paper, we prove that $I(D)$ satisfies the Ratliff (strong persistence) property in the following three cases: (i) $D$ has an outward leaf; (ii) $D$ has an inward leaf $(u,v)\in E(D)$, where $v$ is a sink vertex; (iii) $D$ has an inward leaf $(u,v)\in E(D)$ with $w(v)=1$. We further show that $(I(D)^2:I(D))=I(D)$ if $D$ contains a vertex with in-degree less than or equal to 1, and $(I(D)^3:I(D))=I(D)^2$ when $D$ is either a weighted oriented cycle, or a tree. Finally, if $D$ contains no source vertex, then any associated prime of $I(D)^k$, other than the irrelevant maximal ideal, is also an associated prime of $I(D)^{k+1}$. In addition, if $D$ contains a vertex of in-degree one and all the vertices of $D$ have non-trivial weights, we show that the persistence property holds.