On the path ideals of chordal graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914039315038208 |
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| author | Das, Kanoy Kumar Roy, Amit Saha, Kamalesh |
| author_facet | Das, Kanoy Kumar Roy, Amit Saha, Kamalesh |
| contents | In this article, we give combinatorial formulas for the regularity and the projective dimension of $3$-path ideals of chordal graphs, extending the well-known formulas for the edge ideals of chordal graphs given in terms of the induced matching number and the big height, respectively. As a consequence, we get that the $3$-path ideal of a chordal graph is Cohen-Macaulay if and only if it is unmixed. Additionally, we show that the Alexander dual of the $3$-path ideal of a tree is vertex splittable, thereby resolving the $t=3$ case of a recent conjecture in [Internat. J. Algebra Comput., 33(3):481--498, 2023]. Also, we give examples of chordal graphs where the duals of their $t$-path ideals are not vertex splittable for $t\ge 3$. Furthermore, we extend the formula of the regularity of $3$-path ideals of chordal graphs to all $t$-path ideals of caterpillar graphs. We then provide some families of graphs to show that these formulas for the regularity and the projective dimension cannot be extended to higher $t$-path ideals of chordal graphs (even in the case of trees). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15897 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the path ideals of chordal graphs Das, Kanoy Kumar Roy, Amit Saha, Kamalesh Combinatorics Commutative Algebra 05E40, 13F55, 05C05 In this article, we give combinatorial formulas for the regularity and the projective dimension of $3$-path ideals of chordal graphs, extending the well-known formulas for the edge ideals of chordal graphs given in terms of the induced matching number and the big height, respectively. As a consequence, we get that the $3$-path ideal of a chordal graph is Cohen-Macaulay if and only if it is unmixed. Additionally, we show that the Alexander dual of the $3$-path ideal of a tree is vertex splittable, thereby resolving the $t=3$ case of a recent conjecture in [Internat. J. Algebra Comput., 33(3):481--498, 2023]. Also, we give examples of chordal graphs where the duals of their $t$-path ideals are not vertex splittable for $t\ge 3$. Furthermore, we extend the formula of the regularity of $3$-path ideals of chordal graphs to all $t$-path ideals of caterpillar graphs. We then provide some families of graphs to show that these formulas for the regularity and the projective dimension cannot be extended to higher $t$-path ideals of chordal graphs (even in the case of trees). |
| title | On the path ideals of chordal graphs |
| topic | Combinatorics Commutative Algebra 05E40, 13F55, 05C05 |
| url | https://arxiv.org/abs/2405.15897 |