Normally torsion-free edge ideals of weighted oriented graphs

Abstract

Let I = I(D) be the edge ideal of a weighted oriented graph D, let G be the underlying graph of D, and let I (n) be the n-th symbolic power of I defined using the associated minimal primes of I. We prove that I 2 = I (2) if and only if every vertex of D with weight greater than 1 is a sink and G has no triangles. As a consequence, using a result of Mandal and Pradhan and the classification of normally torsion-free edge ideals of graphs, it follows that I (n) = I n for all n ≥ 1 if and only if every vertex of D with weight greater than 1 is a sink and G is bipartite. If I has no embedded primes, these two properties classify when I is normally torsion-free.

Symbolic powers of edge ideals of weighted oriented graphs

“…Using Macaulay 2, the strong vertex covers of D are {x 1 , x 3 }, {x 2 , x 3 } and {x 2 , x 4 }. Note that {x 2 , x 4 } is a maximal strong vertex cover, but I(D) ⊈ (x 7 2 , x 4 ) because x 2 x 3 ∉ (x 7 2 , x 4 ).…”

“…This ideal was first studied in [6,10]. Recently in [7], the authors give the classification of some normally torsion-free edge ideals of weighted oriented graphs, where the s−th symbolic power of I defined using the associated minimal primes of I. The strong vertex covers of weighted oriented graphs always play an important role in the study of symbolic powers of their edge ideals.…”