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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2407.15206 |
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| _version_ | 1866909758674436096 |
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| author | Muñoz-George, Humberto Reyes, Enrique Villarreal, Rafael H. |
| author_facet | Muñoz-George, Humberto Reyes, Enrique Villarreal, Rafael H. |
| contents | Let $G$ be a graph and let $J=I_c(G)$ be its ideal of covers. The aims of this work are to study the {\rm v}-number ${\rm v}(J)$ of $J$ and to study when $J$ is linearly presented using combinatorics and commutative algebra. We classify when ${\rm v}(J)$ attains its minimum and maximum possible values in terms of the vertex covers of the graph that satisfy the exchange property. If the cover ideal of a graph has a linear presentation, we express its v-number in terms of the covering number of the graph. If $G$ is unmixed, the graph $\mathcal{G}_J$ of $J$ is the graph whose vertices are the minimal vertex covers of $G$ and whose edges are the pairs $\{C,C'\}$ such that $|C\cup C'|=|C|+1$. We show necessary and sufficient conditions for the graph $\mathcal{G}_J$ of $J$ to be connected. Then, for unmixed König graphs, we classify when $J$ is linearly presented using graph theory, and show some results on Cohen--Macaulay König graphs. If $G$ is unmixed, it is shown that the columns of the linear syzygy matrix of $J$ are linearly independent if and only if $\mathcal{G}_J$ has no strong $3$-cycles. One of our main theorems shows that if $G$ is unmixed and has no induced $4$-cycles, then $J$ is linearly presented. For unmixed graphs without $3$- and $5$-cycles, we classify combinatorially when $J$ is linearly presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15206 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The v-numbers and linear presentations of ideals of covers of graphs Muñoz-George, Humberto Reyes, Enrique Villarreal, Rafael H. Commutative Algebra Combinatorics 13F20, 05C22 Let $G$ be a graph and let $J=I_c(G)$ be its ideal of covers. The aims of this work are to study the {\rm v}-number ${\rm v}(J)$ of $J$ and to study when $J$ is linearly presented using combinatorics and commutative algebra. We classify when ${\rm v}(J)$ attains its minimum and maximum possible values in terms of the vertex covers of the graph that satisfy the exchange property. If the cover ideal of a graph has a linear presentation, we express its v-number in terms of the covering number of the graph. If $G$ is unmixed, the graph $\mathcal{G}_J$ of $J$ is the graph whose vertices are the minimal vertex covers of $G$ and whose edges are the pairs $\{C,C'\}$ such that $|C\cup C'|=|C|+1$. We show necessary and sufficient conditions for the graph $\mathcal{G}_J$ of $J$ to be connected. Then, for unmixed König graphs, we classify when $J$ is linearly presented using graph theory, and show some results on Cohen--Macaulay König graphs. If $G$ is unmixed, it is shown that the columns of the linear syzygy matrix of $J$ are linearly independent if and only if $\mathcal{G}_J$ has no strong $3$-cycles. One of our main theorems shows that if $G$ is unmixed and has no induced $4$-cycles, then $J$ is linearly presented. For unmixed graphs without $3$- and $5$-cycles, we classify combinatorially when $J$ is linearly presented. |
| title | The v-numbers and linear presentations of ideals of covers of graphs |
| topic | Commutative Algebra Combinatorics 13F20, 05C22 |
| url | https://arxiv.org/abs/2407.15206 |