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On the path ideals of chordal graphs
- Publication Year :
- 2024
-
Abstract
- 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).<br />Comment: 22 pages. Comments are welcome
- Subjects :
- Mathematics - Combinatorics
Mathematics - Commutative Algebra
05E40, 13F55, 05C05
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2405.15897
- Document Type :
- Working Paper