Properties of symbolic powers of edge ideals of weighted oriented graphs

TL;DR

This paper studies the properties of symbolic powers of edge ideals in weighted oriented graphs, providing methods to find generators, conditions for equality of symbolic and ordinary powers, and characterizations for specific graph classes.

Contribution

It introduces a method to find minimal generators of certain intersections of irreducible ideals and characterizes when symbolic and ordinary powers coincide in weighted rooted trees.

Findings

Method to determine minimal generators of $I_{ ext{subseteq C}}$

Conditions under which symbolic and ordinary powers differ

Characterization of weighted rooted trees with equal symbolic and ordinary powers

Abstract

Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. We provide one method to find all the minimal generators of $ I_{\subseteq C} $, where $ C $ is a maximal strong vertex cover of $D$ and $ I_{\subseteq C} $ is the intersections of irreducible ideals associated to the strong vertex covers contained in $C$. If $ D^{\prime} $ is an induced digraph of $D$, under certain condition on the strong vertex covers of $ D^{\prime} $ and $D$, we show that $ {I(D^{\prime})}^{(s)} \neq {I(D^{\prime})}^s $ for some $s \geq 2$ implies $ {I(D)}^{(s)} \neq {I(D)}^s $. We characterize all the maximal strong vertex covers of $D$ such that at most one edge is oriented into each of its vertex and $w(x) \geq 2$ if $\deg_D(x)\geq 2 $ for all $x \in V(D)$. If $ D $ is a weighted rooted tree with degree of root is $ 1 $ and $ w(x) \geq 2 $ when $ \deg_D(x) \geq 2 $ for all $ x \in V(D) $, we show that $ {I(D)}^{(s)} = {I(D)}^s $ for all $s \geq 2$