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arXiv · 2309.16810

Componentwise linearity of edge ideals of weighted oriented graphs

Manohar Kumar·Ramakrishna Nanduri·Kamalesh Saha
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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 $I(D)$: $$ \text{Vertex splittable}\,\, \Longleftrightarrow\,\, \text{Linear quotient}\,\, \Longleftrightarrow\,\, \text{Componentwise linear}.$$

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Manohar Kumar, Ramakrishna Nanduri, Kamalesh Saha. 2023-09-28. Componentwise linearity of edge ideals of weighted oriented graphs. https://arxiv.org/abs/2309.16810

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