Case-free characterization for d at least 4

Characterize all graphs in A_d for d≥4 without using a case-by-case analysis.

Background

The paper obtains complete classifications of disconnected one-point unions of cycles in A_d for d=1,2,3 through extensive case-by-case algebraic analyses. The authors note that the same general approach can yield necessary and sufficient conditions for larger d, but the resulting analysis becomes increasingly complex.

This open problem asks for a characterization of all graphs in A_d when d≥4 that avoids such case-by-case treatment. The requested result concerns the structure of disconnected one-point unions of cycles admitting local antimagic chromatic number 2.

References

Problem 4.2. For d ≥ 4, characterize all graphs in A_d without a case-by-case analysis.

— Complete characterization of disconnected one-point unions of cycles with local antimagic chromatic number 2  (2610.22808 - Lau et al., 16 Oct 2026) in Problem 4.2, Section 4, p. 24

Conjecture 4.2. As d → ∞, A_d has infinitely many distinct families of solutions (with or without all isomorphic components).

— Complete characterization of disconnected one-point unions of cycles with local antimagic chromatic number 2  (2610.22808 - Lau et al., 16 Oct 2026) in Conjecture 4.2, Section 4, p. 24