Ordinary and symbolic powers of edge ideals of weighted oriented graphs

Abstract

Let D be a weighted oriented graph and I(D) be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of I(D) coincide when D is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines.

Symbolic powers of edge ideals of weighted oriented graphs

“…In [8], we see that, if the set of all vertices of a weighted oriented graph forms a strong vertex cover, all the ordinary and symbolic powers of its edge ideal coincide. Comparision of ordinary and symbolic powers has been done for several classes of weighted oriented graphs in [1], [2] and [9]. In all those papers, to compute the symbolic powers, the authors always find the minimal generators of the intersections of irreducible ideals associated to the strong vertex covers contained in a maximal strong vertex cover.…”

“…In Theorem 5.5, we prove the converse of Lemma 5.2 is also true under the assumption " at most one edge is oriented into each vertex of D and w(x) ≥ 2 if deg D (x) ≥ 2 for all x ∈ V (D)". Recently in [1], Banerjee et al prove the equality of ordinary and symbolic powers of certain class of weighted rooted trees. In this paper, we have extended that result.…”