2024/05/24 by Kanoy Kumar Das, Das, Kanoy Kumar, Amit Roy +3 · 1 citation
Mathematics · Medicine · #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #Cholinesterase and Neurodegenerative Diseases
paper · pdf · doi:10.48550/arxiv.2405.15897
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≥ 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).