Associated primes of powers of connected circulant graph ideals

Determine whether, for every connected circulant graph G=C(n,S) with |S|≥2, the maximal homogeneous ideal is an associated prime of S/NI(G)^t for every integer t≥n/|S|.

Background

The paper considers closed neighborhood ideals NI(G) associated with graphs and studies when the maximal homogeneous ideal becomes an associated prime of a power of such an ideal. A preceding proposition establishes the asserted behavior for a specific two-generator circulant graph under the weaker threshold t≥n/2. The authors then formulate a conjecture extending this phenomenon to every connected circulant graph with at least two generators, with threshold n/|S|.

References

We conjecture that Assume that $G=C(n,S)$ is a connected circulant graph. Asume that $|S| \ge 2$. Then $$ is an associated prime of $S/NI(G)t$ for all $t \ge n/|S|$.

— Stabilization index of V-number of powers of edge ideals of graphs  (2609.09632 - Hien et al., 9 Sep 2026) in Conjecture 0 (labelled \texttt{conj0})