Componentwise linearity of edge ideals of weighted oriented graphs

TL;DR

This paper investigates the conditions under which edge ideals of weighted oriented graphs are componentwise linear, establishing connections with graph properties like co-chordality and providing combinatorial characterizations.

Contribution

It extends Fr"oberg's theorem to weighted oriented graphs and offers new combinatorial criteria for componentwise linearity.

Findings

If the edge ideal is componentwise linear, then the underlying simple graph is co-chordal.

Provides combinatorial characterizations when certain vertex conditions are met.

Shows equivalence of vertex splittable, linear quotient, and componentwise linear for specific graph classes.

Abstract

In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if $D$ is a weighted oriented graph whose edge ideal $I(D)$ is componentwise linear, then the underlying simple graph $G$ of $D$ is co-chordal. This is an analogue of Fr\"oberg's theorem for weighted oriented graphs. We give combinatorial characterizations of componentwise linearity of $I(D)$ if $V^+$ are sinks or $\vert V^+ \vert\leq 1$. Furthermore, if $G$ is chordal or bipartite or $V^+$ are sinks or $\vert V^+ \vert\leq 1$, then we show the following equivalence for $I(D)$: $$ \text{Vertex splittable}\,\, \Longleftrightarrow\,\, \text{Linear quotient}\,\, \Longleftrightarrow\,\, \text{Componentwise linear}.$$