Asymptotic Growth of Associated Primes of Certain Graph Ideals

Abstract

We specify a class of graphs, H _t , and characterize the irreducible decompositions 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 ≥ 0, does there exist a (hyper) graph H _t such that stabilization of associated primes occurs at n ≥ (χ(H _t ) −1) + t?” [4]. For each t, H _t has chromatic number 3 and associated primes that stabilize at n = 2 + t.

Key Words:

• Associated primes
• Chromatic number
• Cover ideals
• Monomial ideals
• Odd cycles

2010 Mathematics Subject Classification:

• 05C25
• 05E40
• 05C38
• 13F20

ACKNOWLEDGMENTS

Many of the results of this article started as computer experiments using the program Macaulay 2 [5]. The author would like to thank Amelia Taylor for introducing her to the subject and for her invaluable support and guidance throughout the research and writing processes, Thomas Shemanske for his help in the editing process, and Adam Van Tuyl for a helpful discussion of Lemma 3.3. We also thank an anonymous referee for their insights and suggestions for improvement.

Notes

Communicated by I. Swanson.