Associated primes of monomial ideals and odd holes in graphs

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Let G be a finite simple graph with edge ideal I(G). Let J(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 J(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 J(G)².