Binomial edge ideals of Cameron-Walker graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918133601665024 |
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| author | Hibi, Takayuki Madani, Sara Saeedi |
| author_facet | Hibi, Takayuki Madani, Sara Saeedi |
| contents | Let $G$ be a Cameron--Walker graph on $n$ vertices and $J_G$ the binomial edge ideal of $G$. Let $S$ denote the polynomial ring in $2n$ variables over a field. It is shown that the following conditions are equivalent: (i) $S/J_G$ is Cohen--Macaulay; (ii) $J_G$ is unmixed; (iii) $\dim (S/J_G) = n+1$; (iv) (a) $n = 3$ and $G$ is a path of length $2$ or (b) $n = 5$ and $G$ is a path of length $4$ or (c) $n=5$ and $G$ is obtained by attaching a path of length $2$ to a triangle. Moreover, the depth of $S/J_G$ is computed for a class of Cameron--Walker graphs, called minimal dense Cameron--Walker graphs. As an application, it is proved that finite graphs $G$ with $\depth(S/J_G)=6$ can have any number of vertices~$n\geq 6$. Finally, it is shown that given integers $t,n$ with $6\leq t\leq n+1$, there exists a finite connected graph $G$ with $\depth (S/J_G)=t$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01150 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Binomial edge ideals of Cameron-Walker graphs Hibi, Takayuki Madani, Sara Saeedi Commutative Algebra Combinatorics 05E40, 13H10 Let $G$ be a Cameron--Walker graph on $n$ vertices and $J_G$ the binomial edge ideal of $G$. Let $S$ denote the polynomial ring in $2n$ variables over a field. It is shown that the following conditions are equivalent: (i) $S/J_G$ is Cohen--Macaulay; (ii) $J_G$ is unmixed; (iii) $\dim (S/J_G) = n+1$; (iv) (a) $n = 3$ and $G$ is a path of length $2$ or (b) $n = 5$ and $G$ is a path of length $4$ or (c) $n=5$ and $G$ is obtained by attaching a path of length $2$ to a triangle. Moreover, the depth of $S/J_G$ is computed for a class of Cameron--Walker graphs, called minimal dense Cameron--Walker graphs. As an application, it is proved that finite graphs $G$ with $\depth(S/J_G)=6$ can have any number of vertices~$n\geq 6$. Finally, it is shown that given integers $t,n$ with $6\leq t\leq n+1$, there exists a finite connected graph $G$ with $\depth (S/J_G)=t$. |
| title | Binomial edge ideals of Cameron-Walker graphs |
| topic | Commutative Algebra Combinatorics 05E40, 13H10 |
| url | https://arxiv.org/abs/2509.01150 |