Associated primes of monomial ideals and odd holes in graphs

Let G be a finite simple graph with edge ideal I(G). Let I(G)∨ denote the Alexander dual of I(G). We show that a description of all induced cycles of odd length in G is encoded in the associated primes of (I(G)∨)2. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simple algebraic criterion for determining whether a graph is perfect. We also show how to determine the existence of odd holes in a graph from the value of the arithmetic degree of (I(G)∨)2.