Comparing symbolic powers of edge ideals of weighted oriented graphs

Abstract

Let D be a weighted oriented graph and I(D) be its edge ideal. We give necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideals of certain classes of weighted oriented graphs. Let D ′ be the weighted oriented graph obtained from D after replacing the weights of vertices with non-trivial weights which are sink, by trivial weights. We show that symbolic powers of I(D) and I(D ′ ) behave in a similar way. Finally, if D is any weighted oriented star graph or some specific weighted naturally oriented path, we show that I(D) (s) = I(D) s for all s ≥ 2.