Regularity of $3$-Path Ideals of Trees and Unicyclic Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916653525106688 |
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| author | Kumar, Rajiv Sarkar, Rajib |
| author_facet | Kumar, Rajiv Sarkar, Rajib |
| contents | Let $G$ be a simple graph and $I_3(G)$ be its $3$-path ideal in the corresponding polynomial ring $R$. In this article, we prove that for an arbitrary graph $G$, $reg(R/I_3(G))$ is bounded below by $2ν_3(G)$, where $ν_3(G)$ denotes the $3$-path induced matching number of $G$. We give a class of graphs, namely, trees for which the lower bound is attained. Also, for a unicyclic graph $G$, we show that $reg(R/I_3(G))\leq 2ν_3(G)+2$ and provide an example that shows that the given upper bound is sharp. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12111 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity of $3$-Path Ideals of Trees and Unicyclic Graphs Kumar, Rajiv Sarkar, Rajib Commutative Algebra Combinatorics 13D02, 13C13, 05E40 Let $G$ be a simple graph and $I_3(G)$ be its $3$-path ideal in the corresponding polynomial ring $R$. In this article, we prove that for an arbitrary graph $G$, $reg(R/I_3(G))$ is bounded below by $2ν_3(G)$, where $ν_3(G)$ denotes the $3$-path induced matching number of $G$. We give a class of graphs, namely, trees for which the lower bound is attained. Also, for a unicyclic graph $G$, we show that $reg(R/I_3(G))\leq 2ν_3(G)+2$ and provide an example that shows that the given upper bound is sharp. |
| title | Regularity of $3$-Path Ideals of Trees and Unicyclic Graphs |
| topic | Commutative Algebra Combinatorics 13D02, 13C13, 05E40 |
| url | https://arxiv.org/abs/2503.12111 |