Associated primes of monomial ideals and odd holes in graphs

Details Associated primes of monomial ideals and odd holes in graphs Associated primes of monomial ideals and odd holes in graphs

2008-06-06

arxiv.org/abs/0806.1159v2

Mathematics

Commutative Algebra

14 pages. In v2, paper has been rewritten and shortened. To appear in J. Algebraic Combin

Scientific paper

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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