D-antimagic orientations of cycles for two-element distance sets

List all orientations of a cycle that admit a D-antimagic labeling when the distance set D has cardinality two.

Background

For distance sets of cardinality two, the paper establishes results for unidirectional odd cycles and gives examples of non-unidirectional oriented cycles that are D-antimagic. These examples show that the known characterization is incomplete, motivating a classification of all cycle orientations in the case ∣D∣=2|D|=2.

References

Although for distance sets of cardinality 2, we only consider unidirectional cycles, we have examples in Figure \ref{fig:C6 D=2} where we provide non-unidirectional oriented $\overrightarrow{C_6}$s that are ${0,3}$-, ${0,2}$-, and ${0,1}$-antimagic. This leads to the following open problem. For $|D|=2$, list all orientations of a cycle to be $D$-antimagic.

— D-Antimagic Labelings of Oriented 2-Regular Graphs  (2501.05123 - Abrar et al., 9 Jan 2025) in Section 2, subsection “D-antimagic unidirectional cycles, with |D|=2,” immediately after the discussion of Figure \ref{fig:C6 D=2}