Stable value of depth of symbolic powers of edge ideals of graphs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914838659203072 |
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| author | Minh, Nguyen Cong Trung, Tran Nam Vu, Thanh |
| author_facet | Minh, Nguyen Cong Trung, Tran Nam Vu, Thanh |
| contents | Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that
$$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le \operatorname{bc}(G),$$
where $I(G)$ denotes the edge ideal of $G$ and $S = \mathrm{k}[x_1, \ldots, x_n]$ is a standard graded polynomial ring over a field $\mathrm{k}$. We further compute the depth of symbolic powers of edge ideals of several classes of graphs, including odd cycles and whisker graphs of complete graphs to illustrate the cases where the above inequality becomes equality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_09967 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stable value of depth of symbolic powers of edge ideals of graphs Minh, Nguyen Cong Trung, Tran Nam Vu, Thanh Commutative Algebra 13D02, 13F55, 05E40 Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that $$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le \operatorname{bc}(G),$$ where $I(G)$ denotes the edge ideal of $G$ and $S = \mathrm{k}[x_1, \ldots, x_n]$ is a standard graded polynomial ring over a field $\mathrm{k}$. We further compute the depth of symbolic powers of edge ideals of several classes of graphs, including odd cycles and whisker graphs of complete graphs to illustrate the cases where the above inequality becomes equality. |
| title | Stable value of depth of symbolic powers of edge ideals of graphs |
| topic | Commutative Algebra 13D02, 13F55, 05E40 |
| url | https://arxiv.org/abs/2308.09967 |