On the path ideals of chordal graphs

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).