Depth and Stanley depth of the path ideal associated to an nn-cyclic graph

Abstract

We compute the depth and Stanley depth for the quotient ring of the path ideal of length 3 associated to a n-cyclic graph, given some precise formulas for the depth when n ̸≡ 1 (mod 4), tight bounds when n ≡ 1 (mod 4), and for Stanley depth when n ≡ 0, 3 (mod 4), tight bounds when n ≡ 1, 2 (mod 4). We also give some formulas for the depth and Stanley depth of a quotient of the path ideals of length n − 1 and n.