Associated primes of monomial ideals and odd holes in graphs

TL;DR

This paper links the combinatorial structure of graphs, specifically odd cycles and perfectness, to algebraic properties of associated monomial ideals, providing algebraic methods to detect odd holes and perfect graphs.

Contribution

It introduces an algebraic approach to identify odd cycles and perfectness in graphs using associated primes and ideal operations on monomial ideals.

Findings

Associated primes of $J(G)^2$ encode all odd induced cycles in $G$.

A simple algebraic criterion determines if a graph is perfect.

The existence of odd holes can be inferred from the arithmetic degree of $J(G)^2$.

Abstract

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 $J(G)^2$.