1. Square-free powers of Cohen-Macaulay forests, cycles, and whiskered cycles
- Author
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Das, Kanoy Kumar, Roy, Amit, and Saha, Kamalesh
- Subjects
Mathematics - Commutative Algebra ,Mathematics - Combinatorics ,Primary: 13C15, 05E40, 13D02, Secondary: 13H10, 05C70 - Abstract
Let $I(G)^{[k]}$ denote the $k^{th}$ square-free power of the edge ideal $I(G)$ of a graph $G$. In this article, we provide a precise formula for the depth of $I(G)^{[k]}$ when $G$ is a Cohen-Macaulay forest. Using this, we show that for a Cohen-Macaualy forest $G$, the $k^{th}$ square-free power of $I(G)$ is always Cohen-Macaulay, which is quite surprising since all ordinary powers of $I(G)$ can never be Cohen-Macaulay unless $G$ is a disjoint union of edges. Additionally, we provide tight bounds for the regularity and depth of $I(G)^{[k]}$ when $G$ is either a cycle or a whiskered cycle, which aids in identifying when such ideals have linear resolution. Furthermore, we provide combinatorial formulas for the depth of second square-free powers of edge ideals of cycles and whiskered cycles. We also obtained an explicit formula of the regularity of second square-free power for whiskered cycles., Comment: Comments are welcome
- Published
- 2024