Uniqueness of extremal graphs for paths

Determine whether the extremal graph constructed for the weak rainbow saturation number of P_ℓ is unique when ℓ ≥ 30.

Background

The paper proves that rwsat(n,P_ℓ)=ℓ+1 for n≥ℓ+1 and ℓ≥31, and constructs an extremal graph attaining this value. It also notes that for small path orders, including ℓ≤8, extremal graphs need not be unique.

For larger path orders, the authors have not determined whether the constructed extremal graph is the only graph attaining the minimum weak rainbow saturation number. The remark formulates this as an unresolved uniqueness question for ℓ≥30.

References

When $\ell\geq 30$, although we have constructed an extremal graph, it remains unknown whether it is unique.

— Weak rainbow saturation numbers of paths, stars and cycles  (2609.03823 - Bo et al., 3 Sep 2026) in Remark following the proof of Theorem main:star, Section 2 (Stars)