Associated primes of monomial ideals and odd holes in graphs

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 $J(G)^2$.

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