Componentwise linearity of powers of edge ideals of weighted oriented graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914044628172800 |
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| author | Kumar, Manohar Mondal, Joydip Nanduri, Ramakrishna |
| author_facet | Kumar, Manohar Mondal, Joydip Nanduri, Ramakrishna |
| contents | In this paper, we study the componentwise linearity of powers of edge ideal of a weighted oriented graph $D$. We give a characterization for componentwise linearity of the edge ideal $I(D)$ in terms of forbidden subgraphs of $D$. If $D$ is house-free or complete $r$-partite, then the following statements are equivalent: (1) $I(D)$ is componentwise linear; (2) $I(D)$ is vertex splittable; (3) $I(D)$ has linear quotient property; (4) both $G$ and $H(I(D)_{(2)})$ are co-chordal and $D_1,D_2,D_3,D_4$ as in Figure 3, are not induced subgraphs of $D$. Furthermore, if $D$ is a complete $r$-partite weighted oriented graph, then we show that: $I(D)^k$ is componentwise linear, for some $k\geq 2 \iff I(D)$ is componentwise linear. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14637 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Componentwise linearity of powers of edge ideals of weighted oriented graphs Kumar, Manohar Mondal, Joydip Nanduri, Ramakrishna Commutative Algebra 13D02, 05E40, 05E99, 13D45 In this paper, we study the componentwise linearity of powers of edge ideal of a weighted oriented graph $D$. We give a characterization for componentwise linearity of the edge ideal $I(D)$ in terms of forbidden subgraphs of $D$. If $D$ is house-free or complete $r$-partite, then the following statements are equivalent: (1) $I(D)$ is componentwise linear; (2) $I(D)$ is vertex splittable; (3) $I(D)$ has linear quotient property; (4) both $G$ and $H(I(D)_{(2)})$ are co-chordal and $D_1,D_2,D_3,D_4$ as in Figure 3, are not induced subgraphs of $D$. Furthermore, if $D$ is a complete $r$-partite weighted oriented graph, then we show that: $I(D)^k$ is componentwise linear, for some $k\geq 2 \iff I(D)$ is componentwise linear. |
| title | Componentwise linearity of powers of edge ideals of weighted oriented graphs |
| topic | Commutative Algebra 13D02, 05E40, 05E99, 13D45 |
| url | https://arxiv.org/abs/2509.14637 |