Asymptotic Growth of Associated Primes of Certain Graph Ideals

TL;DR

This paper investigates the asymptotic behavior of associated primes in powers of cover ideals for a specific class of graphs, revealing stabilization patterns tied to graph chromatic number and answering a posed question.

Contribution

It characterizes the irreducible decomposition of all powers of cover ideals for a class of graphs and demonstrates the stabilization of associated primes at predictable thresholds.

Findings

Associated primes stabilize at s=2+t for graphs H_t.

Graphs H_t have chromatic number 3.

Provides an affirmative answer to a question about prime stabilization thresholds.

Abstract

We specify a class of graphs, $H_t$, and characterize the irreducible decomposition of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question "For each integer $t\geq 0$, does there exist a (hyper) graph $H_t$ such that stabilization of associated primes occurs at $s\geq (\chi(H_t)-1)+t$?" asked by Francisco, H\`a, and Van Tuyl. For each $t$, $H_t$ has chromatic number 3 and associated primes that stabilize at $s=2+t$.