Asymptotic Growth of Associated Primes of Certain Graph Ideals

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 $t\ge 0$, does there exist a (hyper) graph $H_t$ such that stabilization of associated primes occurs at $s\ge (\chi (H_t)-1)+t$?" asked by Francisco, Hà, and Van Tuyl. For each $t$, $H_t$ has chromatic number 3 and associated primes that stabilize at $s=2+t$.

14 pages, 1 figure. In v2, paper had been rewritten. To appear in Communications in Algebra