Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917418074374144 |
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| author | Das, Sanjoy Ghosh, Arka Selvaraja, S |
| author_facet | Das, Sanjoy Ghosh, Arka Selvaraja, S |
| contents | Let $G$ be a finite simple graph and let $I(G)$ denote its edge ideal. For $q \ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by squarefree monomials corresponding to matchings of size $q$ in $G$. We denote by $\operatorname{reg}(-)$ the Castelnuovo-Mumford regularity. Das, Roy, and Saha conjectured that if $G = W(C_n)$ is a whiskered cycle, then \[ \operatorname{reg}\big(I(G)^{[q]}\big) = 2q + \left\lfloor \frac{n - q - 1}{2} \right\rfloor ~ \text{for all } 1 \le q \le ν(G), \] where $ν(G)$ denotes the matching number of $G$. In this paper, we confirm this conjecture by determining the exact value of $\operatorname{reg}(I(G)^{[q]})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles Das, Sanjoy Ghosh, Arka Selvaraja, S Commutative Algebra 13D02, 05E40, 05C70, 13F55 Let $G$ be a finite simple graph and let $I(G)$ denote its edge ideal. For $q \ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by squarefree monomials corresponding to matchings of size $q$ in $G$. We denote by $\operatorname{reg}(-)$ the Castelnuovo-Mumford regularity. Das, Roy, and Saha conjectured that if $G = W(C_n)$ is a whiskered cycle, then \[ \operatorname{reg}\big(I(G)^{[q]}\big) = 2q + \left\lfloor \frac{n - q - 1}{2} \right\rfloor ~ \text{for all } 1 \le q \le ν(G), \] where $ν(G)$ denotes the matching number of $G$. In this paper, we confirm this conjecture by determining the exact value of $\operatorname{reg}(I(G)^{[q]})$. |
| title | Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles |
| topic | Commutative Algebra 13D02, 05E40, 05C70, 13F55 |
| url | https://arxiv.org/abs/2604.17100 |