Asymptotic Growth of Associated Primes of Certain Graph Ideals

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 $s=2+t$.