Orientations of cycles with zero minimum distance

List all orientations of an oriented cycle that admit a D-antimagic labeling when the minimum element of the distance set D is zero.

Background

The paper proves that if an oriented cycle is D-antimagic and its orientation is neither unidirectional nor Θ-oriented, then the minimum distance in D must be zero. The authors do not determine which orientations can actually be D-antimagic in this remaining case, leaving a classification problem for distance sets containing zero.

References

However, we could not identify all possible orientations for a $D$-antimagic oriented cycle when $\min(D)=0$. Thus, the following open problem: For $\min(D)=0$, list all possible orientations such that the oriented cycle $\overrightarrow{C_n}$ is $D$-antimagic.

— D-Antimagic Labelings of Oriented 2-Regular Graphs  (2501.05123 - Abrar et al., 9 Jan 2025) in Section 2, immediately after Lemma 2.5 (the lemma labeled lem:min0)