Let G be a simple graph and \(I_t(G)\) denote the t-path ideal of G. It is well known that the Castelnuovo–Mumford regularity \(\textrm{reg}(R/I_t(G))\) and the projective dimension \(\textrm{pd}(R/I_t(G))\) are bounded below by the quantities \((t-1)\nu _t(G)\) and the big height \(\textrm{bight}(I_t(G))\) , respectively, where \(\nu _t(G)\) denotes induced matching number of the hypergraph corresponding to \(I_t(G)\) . We show that if \(t\ge 4\) , then the difference between \(\textrm{reg}(R/I_t(G))\) and \((t-1)\nu _t(G)\) , and the difference between \(\textrm{pd}(R/I_t(G))\) and \(\textrm{bight}(I_t(G))\) can be arbitrarily large even if we take G to be a tree. This, in particular, disproves a conjecture in Hang and Vu (Graphs Combin 41(1):18, 2025). However, when \(t=3\) and G is chordal, we show that \(\textrm{reg}(R/I_3(G))=2\nu _3(G)\) and \(\textrm{pd}(R/I_3(G))=\textrm{bight}(I_3(G))\) , extending the well-known formulas for the edge ideals of chordal graphs. 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 \(I_3(G)\) is vertex splittable when G is a tree, thereby resolving the \(t=3\) case of a recent conjecture in Abdelmalek et al. (Int J Algebra Comput 33(3):481–498, 2023). Also, for each \(t\ge 3\) , we give examples of chordal graphs G such that the duals of the corresponding t-path ideals are not vertex splittable. Furthermore, we extend the formula of the regularity of 3-path ideals of chordal graphs to all t-path ideals of caterpillar graphs.
