Associated primes of monomial ideals and odd holes in graphs

作者：Francisco Christopher A^*; Ha Huy Tai; Van Tuyl Adam

来源：Journal of Algebraic Combinatorics, 2010, 32(2): 287-301.

DOI：10.1007/s10801-010-0215-y

• Edge ideals
• Odd cycles
• Perfect graphs
• Associated primes
• Arithmetic degree

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

• 出版日期2010-9