On the path ideals of chordal graphs

TL;DR

This paper provides combinatorial formulas for the algebraic invariants of 3-path ideals of chordal graphs, explores their Cohen-Macaulay property, and extends regularity results to t-path ideals of caterpillar graphs, addressing conjectures and counterexamples.

Contribution

It introduces explicit formulas for regularity and projective dimension of 3-path ideals of chordal graphs, proves the Cohen-Macaulay criterion, and extends regularity results to t-path ideals of caterpillar graphs.

Findings

3-path ideal of a chordal graph is Cohen-Macaulay iff unmixed.

Alexander dual of a 3-path ideal of a tree is vertex splittable.

Formulas for regularity extend to all t-path ideals of caterpillar graphs.

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).