Factor-critical graphs and dstab, astab for an edge ideal
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911930529087488 |
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| author | Morales, Marcel Dung, Nguyen Thi |
| author_facet | Morales, Marcel Dung, Nguyen Thi |
| contents | Let $G$ be a simple, connected non bipartite graph and let $I_G$ be the edge idealof $G$. In our previous work we showed that L. Lovász's theorem on ear decompositions offactor-critical graphs and the canonical decomposition of a graph given by Edmonds and Gallai are basic tools for the irreducible decomposition of $I^{k}_G$. In this paper we use some tools from graph theory, mainly Withney's theorem on ear decompositions of 2-edge connected graphs in order to introduce a new method to make a graph factor-critical. We can describe the set $\cup_ {k=1}^{\infty}{\rm Ass} (I^{k}_G)$ in terms of some subsets of $G$. We give explicit formulas for the numbers astab$(I_G)$ and dstab$(I_G)$, which are, respectively, the smallest number $k$ such that ${\rm Ass} (I^{k}_G)= {\rm Ass} (I^{k+i}_G)$ for all $i\geq 0$ and the smallest number $k$ such that the maximal ideal belongs to ${\rm Ass}(I^{k}_G)$. We also give very simple upper bounds for astab$(I_G)$ and dstab$(I_G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Factor-critical graphs and dstab, astab for an edge ideal Morales, Marcel Dung, Nguyen Thi Commutative Algebra Let $G$ be a simple, connected non bipartite graph and let $I_G$ be the edge idealof $G$. In our previous work we showed that L. Lovász's theorem on ear decompositions offactor-critical graphs and the canonical decomposition of a graph given by Edmonds and Gallai are basic tools for the irreducible decomposition of $I^{k}_G$. In this paper we use some tools from graph theory, mainly Withney's theorem on ear decompositions of 2-edge connected graphs in order to introduce a new method to make a graph factor-critical. We can describe the set $\cup_ {k=1}^{\infty}{\rm Ass} (I^{k}_G)$ in terms of some subsets of $G$. We give explicit formulas for the numbers astab$(I_G)$ and dstab$(I_G)$, which are, respectively, the smallest number $k$ such that ${\rm Ass} (I^{k}_G)= {\rm Ass} (I^{k+i}_G)$ for all $i\geq 0$ and the smallest number $k$ such that the maximal ideal belongs to ${\rm Ass}(I^{k}_G)$. We also give very simple upper bounds for astab$(I_G)$ and dstab$(I_G)$. |
| title | Factor-critical graphs and dstab, astab for an edge ideal |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2406.16432 |