  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 depth when $n\not\equiv 1\,(\mbox{mod}\ 4)$, tight bounds when
$n\equiv 1\,(\mbox{mod}\ 4)$ and for Stanley depth when $n\equiv
0,3\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1,2\,(\mbox{mod}\ 4)$. Also,
we give some formulas for depth and Stanley depth of a quotient of the path
ideals of length $n-1$ and $n$.
