Cohen-Macaulay and Gorenstein path ideals of trees (Q2822084)

The paper under review is related to algebraic properties of path ideals of trees. In fact, for an integer \(t\geq 2\), the authors introduced a new property ``fitting \(t\)-partitioned'' for rooted directed trees. This property leads them to characterizing unmixedness of the path ideal of a tree. Hence if \(k\) is a field, \(\Gamma\) is a tree over the vertex set \(\{x_1, \dots, x_n\}\), \(2\leq t \leq n\) and \(r\geq 2\), then the authors can show that the unmixedness, Cohen-Macaulayness and satisfying the Serre's condition \(S_r\) of the ring \(k[x_1, \dots, x_n]/I_t(\Gamma)\) are equivalent, where \(I_t(\Gamma)\) is the path ideal of \(\Gamma\). Also, they can find some equivalent conditions for Gorensteinness of this ring. All of the results are new, nice and well-written and the new definitions is very well illuminated by several different examples.