COHEN-MACAULAY PATH IDEALS

Let R = k[x1; : : : ; xn], where k is a field. The path ideal (of length t ¸ 2) of a directed graph G is the monomial ideal, denoted by It(G), whose generators correspond to the directed paths of length t in G. Let Cn be an n-cycle. We prove that for t ¸ 3, It(Cn) is unmixed if and only if t · n · b3t=2c + 1 or n = 2t + 1. Also, we show that R=It(Cn) is Cohen-Macaulay if and only if n = t or t + 1 or 2t + 1.